Pacific Journal of Mathematics Vol 227 Issue 2, Oct 2006

Volume 227

No. 2

October 2006

Pacific Journal of Mathematics

PACIFIC JOURNAL OF MATHEMATICS

Pacific Pacific Journal Journal of of Mathematics Mathematics

2006 Vol. 227, No. 2 Volume 227

No. 2

October 2006

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PACIFIC JOURNAL OF MATHEMATICS Vol. 227, No. 2, 2006

SELF-SIMILAR SOLUTIONS OF THE p-LAPLACE HEAT EQUATION: THE FAST DIFFUSION CASE M ARIE F RANÇOISE B IDAUT-V ÉRON We study the self-similar solutions of the equation u t − div(|∇u| p−2 ∇u) = 0 in R N , where N ≥ 1, p ∈ (1, 2). We provide a complete description of the signed solutions of the form u(x, t) = (±t)−α/β w((±t)−1/β |x|), regular or singular at x = 0, with α, β real, β 6= 0, and possibly not defined on all of R N × (0, ±∞).

1. Introduction and main results In this article we study the existence of self-similar solutions of the degenerate parabolic equation involving the p-Laplace operator in R N , N ≥ 1, (E u )

u t − div(|∇u| p−2 ∇u) = 0,

with 1 < p < 2. In the sequel we set δ=

p , 2− p

so δ > 1. Two critical values P1 , P2 are involved in the problem P1 =

2N , N +1

P2 =

2N ; N +2

see [DiBenedetto and Herrero 1990], for example. They are connected with δ through the relations p > P1 ⇐⇒ δ > N ,

p > P2 ⇐⇒ δ >

N . 2

If u(x, t) is a solution and α, β ∈ R, then u λ (x, t) = λα u(λx, λβ t) is a solution of (E u ) if and only if β = p − (2 − p)α = (2 − p)(δ − α); MSC2000: primary 35K65; secondary 34C35. Keywords: degenerate parabolic equations, self-similar solutions. 201

202

MARIE FRANÇOISE BIDAUT-VÉRON

thus β > 0 if and only if α < δ. For given α ∈ R such that α 6= δ, the natural way to construct particular solutions is to search for self-similar solutions, radially symmetric in x, of the form (1–1)

u = u(x, t) = (εβt)−α/β w(r ),

r = (εβt)−1/β |x|,

where ε = ±1. By translation, for any real T , we obtain solutions defined for any t > T when εβ > 0, or t < T when εβ < 0. The hypersurfaces {r = constant} are of focusing type if β > 0 and of spreading type if β < 0. We are led to the equation (E w )

(|w 0 | p−2 w 0 )0 +

N −1 0 p−2 0 |w | w + ε(r w0 + αw) = 0 r

in(0, ∞).

If we look for solutions of (E u ) under the form u = Ae−εµt w(r ),

r = Me−εµt/δ |x|,

µ > 0,

then w solves (E w ) provided M = δ/α and A = (δ p /α p−1 µ)1/(2− p) , where α > 0 is arbitrary. This is another motivation for studying equation (E w ) for any real α. In the huge literature on self-similar solutions of parabolic equations, many results deal with positive solutions u defined and smooth on R N × (0, ∞). Equation (E w ) was studied in [Qi and Wang 1999] when α > 0, ε = 1. In our work we provide an exhaustive description of the self-similar solutions of equation (E u ), possibly not defined on all of (0, ∞), with constant or changing sign. In particular, for suitable values of α, we prove the existence of solutions w oscillating with respect to 0 as r tends to 0 or ∞, or constant-sign solutions oscillating with respect to some nonzero constant. Our main tool is the reduction of the problem to an autonomous system with two variables and two parameters, p and α. We are led to a dynamical system, which we study by phase-plane techniques. When p = 32 , this system is nearly quadratic, and many devices from the theory of algebraic dynamical systems can be used. In the general case such structures do not exist; then we use energy functions associated to the system. The behavior of the solutions presents great diversity, according to the possible values of p and α. In the sequel we set N−p ; p−1 thus η > 0 if N ≥ 2, and η = −1 if N = 1. Observe the relation connecting η, δ and N : N −η δ−N (1–2) = δ−η = . p−1 2− p η=

Explicit solutions. Obviously if w is a solution of (E w ), so is −w. Many particular solutions are well-known.

SELF-SIMILAR SOLUTIONS OF THE

p -LAPLACE EQUATION

203

The infinite point source solution U∞ . The simplest positive solutions of equation (E w ), which exist for any α such that ε(δ − N )(δ − α) > 0, are given by w(r ) = `r −δ ,

(1–3) where (1–4)

δ− N 1/(2− p) ` = εδ p−1 > 0. δ−α

They correspond to a unique solution u of (E u ) called U∞ in [Chasseigne and Vazquez 2002], singular at x = 0, for any t 6= 0: Ct 1/(2− p) U∞ (x, t) = , C = (2 − p)δ p−1 (δ − N ). |x| p The case α = N. Here the equation (E w ) has a first integral w + εr −1 |w 0 | p−2 w 0 = Cr −N .

(1–5)

All the solutions corresponding to C = 0 are given by 0

(1–6)

0

w = w K ,ε (r ) = ±(εδ −1r p + K )−δ/ p , u = ±u K ,ε (x, t) = (εβ N t)−N /β N (εδ −1 (εβ N t)− p /β N |x| p + K )−( p−1)/(2− p) , K ∈ R, 0

0

with β = β N = (N + 1)( p − P1 ). For p > P1 , ε = 1, K > 0, the solutions are named after Barenblatt [1952]. For given c > 0, the function u K ,1 , defined on R N × (0, ∞), is the unique solution of equation (E u ) withRinitial data u(0) = cδ0 , δ0 being the Dirac mass at 0 and K begin determined by R N u K (x, t) dt = c; see for example [Zhao 1993]. Moreover the functions u K ,1 , with K > 0, are the only nonnegative solutions defined on R N ×(0, ∞), such that u(x, 0) = 0 for any x 6= 0; see [Kamin and Vázquez 1992]. In the case K = 0, we find again the function U∞ , and U∞ is the limit of the functions u K ,1 as K → 0, or equivalently c → ∞. The case α = η. We exhibit a family of solutions of (E w ): (1–7)

w(r ) = Cr −η ,

u(t, x) = C|x|−η = C|x|( p−N )/( p−1) ,

C 6= 0,

Solutions u, independent of t, are the fundamental p-harmonic solutions of the equation when p > P1 . The case α = − p0 . Equation (E w ) admits solutions of the form 0

(1–8)

w(r ) = ±K (N (K p 0 ) p−2 + εr p ), 0

u(x, t) = ±K (N (K p 0 ) p−2 t + ε|x| p ),

K > 0,

204


and the functions u are solutions of the form ψ(t) + 8(|x|) with 8 nonconstant. They have constant sign when ε = 1, and a changing sign when ε = −1. The case α = 0. Here equation (E w ) can be explicitly solved: either w0 ≡ 0 (hence w ≡ a ∈ R, and u is a constant solution of (E u )), or there exists K ∈ R such that  −1/(2− p) ε   r N −η if δ 6= N ,  K+ δ− N (1–9) |w 0 | = r (1−N )/( p−1) × −1/(2− p)  2− p  (K + ε ln r ) if δ = N ,  p−1 which gives w by integration, up to a constant, and then u(x, t) = w(|x|/(εpt)1/ p ). The case N = 1 and α = ( p − 1)/(2 − p) > 0. Here again we obtain explicit solutions: −α −α w(r ) = ± εK (r −(K α) p−1 ) , u(x, t) = ± εK (|x|−ε(K α) p−1 t) , K > 0. All the functions w above are defined on intervals of the form (R, 0), R ≥ 0 if ε = 1, and (0, S), S ≤ ∞ if ε = −1. Note. When α = δ, equation (E u ) is invariant under the transformation u λ (x, t) = λα u(λx, t); searching solutions of the form u(x, t) = |x|−δ ψ(t), we find again the function U∞ . Different kinds of singularities. Consider equation (E w ). It is easy to get local existence and uniqueness near any point r1 > 0; thus any solution w is defined on a maximal interval (Rw , Sw ), with 0 ≤ Rw < Sw ≤ ∞; and in fact Sw = ∞ when ε = 1, and Rw = 0 when ε = −1 (see Theorem 2.2). Returning to solution the u of (E u ) associated to w by (1–1), it is defined on a subset of R N \ {0} × (0, ±∞): Dw = {(x, t) : x ∈ R N , εβt > 0, (εβt)1/β Rw < |x| < (εβt)1/β Sw }. When w is defined on (0, ∞), then u is defined on R N \ {0} × (0, ±∞). Regular solutions. Among the solutions of (E w ) defined near 0, we also show the existence and uniqueness of solutions w = w( . , a) ∈ C 2 ([0, Sw )) such that, for some a ∈ R, (1–10)

w(0) = a,

w0 (0) = 0.

These are called regular solutions. Obviously, they are defined on [0, ∞) when ε = 1. If w is regular, then Dw = R N × (0, ±∞), and u( . , t) ∈ C 1 (R N ) for t 6= 0; we will say that u is regular. This does not imply the regularity up to t = 0: indeed u presents a singularity at time t = 0 if and only if 0 < α < δ. In the sequel we shall not mention the trivial solution w ≡ 0, corresponding to a = 0.


p -LAPLACE EQUATION

205

Singular solutions. If Rw = 0 and w is not regular, u presents a singularity at x = 0 for t 6= 0, called a standing singularity. Following [Vazquez and Véron 1996; Chasseigne and Vazquez 2002], for such a solution, we say that x = 0 is a weak singularity if x 7→ w(|x|) ∈ L 1loc (R N ), or equivalently if u( . , t) ∈ L 1loc (R N ) for t 6= 0; and a strong singularity if not. If u has a strong/weak singularity, and limt→0 u(t, x) = 0 for any x 6= 0, we call u a strong/weak razor blade. If u( . , t) ∈ L 1 (R N ) for t 6= 0, then u is called integrable. Solutions with a reduced domain. If Rw > 0 or Sw < ∞, we say that u and w have a reduced domain. Then Dw has a lateral boundary of the form 6w = {|x| = C(εβt)1/β }, of parabolic type if β > 0 and of hyperbolic type if β < 0, and u has an explosion near 6w . In Proposition 2.15 we calculate the blow-up rate, which is of the order of d(x, t)−( p−1)/(2− p) , where d(x, t) is the distance to 6w . Main results. We give a summary of our main results, expressed in terms of the function u, avoiding for simplicity particular cases (such as N = 1, or α = δ, or p = P1 ) and solutions with a reduced domain (although there exist many such). All cases omitted here and detailed statements in terms of w can be found inside each section. An important critical value of α is given by α∗ = δ +

(1–11)

δ(N − δ) ; ( p − 1)(2δ − N )

it appears when ε = 1, p > P2 , and then α ∗ > 0, or ε = −1, p < P2 , and then α ∗ < 0. Note. To return from w to u, consider any solution w of (E w ) defined on (0, ∞), such that for some λ ≥ 0 and µ ∈ R, lim r →0 r λ w = c 6= 0 and lim r →0 r µ w = c0 6= 0. Then: (i) For fixed t, u has a singularity in |x|−λ near x = 0, and a behavior in |x|−µ for large |x|. Thus x = 0 is a weak singularity if and only if λ < N , and u is integrable if and only if λ < N < µ. (ii) For fixed x 6= 0, the behavior of u near t = 0, depends on the sign of β: lim |x|µ |t|(α−µ)/β u(x, t) = C 6= 0

ifα < δ,

lim |x|λ |t|(α−λ)/β u(x, t) = C 6= 0

ifδ < α.

t→0

t→0

Solutions defined for t > 0. Here we look for solutions u of (E u ) on R N \ {0} × (0, ∞) of the form (1–1). That means εβ > 0, or equivalently ε = 1 and α < δ (see Section 3) or ε = −1, δ < α (see Section 4). We begin with the case ε = 1, and examine the dependence on the sign of p − P1 . For proofs, see Theorems 3.2, 3.4 and 3.5.

206


Theorem 1.1. Assume ε = 1, −∞ < α < δ, p > P1 , and N ≥ 2. Then U∞ is a solution on R N \ {0} × (0, ∞) and a strong razor blade. There exist also positive solutions having a strong singularity in |x|−δ and satisfying limt→0 |x|α u = L > 0 (for x 6= 0). For α ≤ N , any function u( . , t) has at most one zero at time t. (1) For α < N , all regular solutions on R N ×(0, ∞) have constant sign, are not integrable, and they are solutions of (E u ) with initial data L|x|−α ∈ L 1loc (R N ). There exist positive integrable razor blades having a singularity in |x|−η . There exist also positive solutions having a weak regularity in |x|−η and satisfying limt→0 |x|α u = L; in particular if α = η, then u ≡ C|x|−η . There exist solutions with one zero and a weak or a strong singularity. (2) For α = N , all regular (Barenblatt) solutions have constant sign and are integrable. There exist solutions with one zero and a weak singularity. (3) For N < α, all regular solutions have at least one zero. If α < α ∗ , any solution has a finite number of zeros. If N < α ∗ , there exists αˇ ∈ (α ∗ , δ) such that if αˇ < α, regular solutions are oscillating around 0 for large |x|, and r δ w is asymptotically periodic in ln r ; and there exists precisely a solution u such that r δ w is periodic in ln r . Theorem 1.2. Assume ε = 1, −∞ < α < δ, and p < P1 . Then all regular solutions on R N × (0, ∞) have constant sign, are not integrable, and are solutions of (E u ) with initial data L|x|−α ∈ L 1loc (R N ). There is no other solution on R N \{0}×(0, ∞). If α > 0, all the solutions w tend to 0 at ∞, whereas if α < 0, some of the solutions are unbounded near ∞. Next we come to the case ε = −1, which is treated in Theorems 4.1 and 4.2. Theorem 1.3. Assume ε = −1, δ < α, p > P1 , and N ≥ 2. There is no regular solution on R N ×(0, ∞). Besides the function U∞ , which is a strong razor blade, there exist positive integrable razor blades having a singularity in |x|−η , and positive solutions having a strong singularity in |x|−α and satisfying limt→0 |x|α u = L. Theorem 1.4. Assume ε = −1, δ < α, p < P1 (N ≥ 2). There is no regular solution on R N ×(0, ∞). There exists a positive solution on R N \{0}×(0, ∞) with a singularity in |x|−α (strong if and only if N ≤ α), and limt→0 |x|α u = L. Note. Weak singularities can occur even if p > P1 . For example, the solutions u(t, x) = C|x|−η = C|x|( p−N )/( p−1) (N ≥ 2) given in (1–7) have a weak singularity. There even exist positive solutions u with a standing singularity, and integrable; see Theorems 1.1 and 1.3. This is not contradictory with the regularizing effect N N L 1loc (R N ) → L ∞ loc (R ), which concerns solutions in (0, ∞) × R . The functions constructed above are solutions in (0, ∞) × R N \ {0}, and the singularity x = 0 is not removable.


p -LAPLACE EQUATION

207

Solutions defined for t < 0. Next we consider the solutions defined for t < 0, and more generally for t < T . They correspond to ε = 1, δ < α (Section 5), or ε = −1, α < δ (Section 6). A main question in that case is the extinction problem: do there exist regular solutions u vanishing identically on R at time T ? Do there exist singular razor blades, vanishing on R N \ {0} at time T ? Are they integrable? One of our most significant results is the existence of two critical values αcrit > 0 (when P2 < p < P1 ) and α crit < 0 (when 1 < p < P2 ), for which the regular solutions u αcrit are positive, integrable, and vanish identically at time 0. Another new phenomena is the existence of positive solutions such that C1U∞ ≤ u ≤ C2U∞ for some C1 , C2 > 0, with a periodicity property, see Theorems 1.6 and 1.8. First assume ε = 1. From Theorems 5.1 when p > P1 and 5.4, 5.6, 5.7 when p < P1 , we deduce: Theorem 1.5. Assume ε = 1, δ < α, p > P1 , with N ≥ 2. Any solution u on R N \ {0} × (0, −∞), in particular the regular ones, is oscillating around 0 for fixed t < 0 and large |x|, and r δ w is asymptotically periodic in ln r . There exists a solution such that r δ w is periodic in ln r . There exist weak integrable razor blades, with a singularity in |x|−η . Theorem 1.6. Assume ε = 1, δ < α, p < P1 . Then U∞ is a solution on R N \ {0} × (0, −∞), and a weak razor blade. (1) If p < P2 , all regular solutions on R N × (0, −∞) have constant sign, are not integrable, and vanish identically at t = 0, with ku( . , t)k L ∞ (R N ) ≤ C|t|α/|β| . All the solutions have a finite number of zeros. (2) For α < η, regular solutions have constant sign, with the same behavior (given by (1–6) if α = N ). There exists a positive solution u, which is not integrable, with a singularity in |x|−α (a strong one if and only if α ≥ N ), and limt→0 |x|α u = L. If α = η, then u(t, x) = C|x|−η is a solution with a strong singularity. (3) If p > P2 , there exists a critical value αcrit such that η < αcrit < α ∗ and the regular solutions u αcrit have constant sign, are integrable, and vanish identically at t = 0, with ku( . , t)k L ∞ (R N ) ≤ C|t|α/|β| . (4) If α ∈ (αcrit , α ∗ ), there exist positive solutions u such that r δ w is periodic in ln r ; thus C 1 U∞ ≤ u ≤ C 2 U∞

for someC1 , C2 > 0.

There exist positive solutions u, with the same bounds, such that r δ w is asymptotically periodic near 0 . There exist positive integrable solutions u such that r δ w is asymptotically periodic near 0.

208


(5) If αcrit < α, all regular solutions are oscillating around 0 for fixed t < 0 and large |x|, and r δ w is asymptotically periodic in ln r . There exist solutions oscillating around 0, such that r δ w is periodic. If α ∗ < α, there exist positive integrable razor blades, with a singularity in |x|−δ . Finally suppose ε = −1. From Theorems 6.1, 6.2 when p > P1 and 6.4, 6.6, 6.8, 6.9 when p < P1 , we obtain: Theorem 1.7. Assume ε = −1, α < δ and p > P1 , with N ≥ 2. If α > 0, there exist positive solutions u with a weak singularity in |x|−η , integrable if and only if α > N , and limt→0 |x|α u = L. If α < 0, any solution has at least a zero. If − p 0 < α, there is no regular solution on R N × (0, −∞). If α = − p 0 , all regular solutions, given by (1–8), have one zero. Theorem 1.8. Assume ε = −1, α < δ and p < P1 . Then U∞ is a solution on R N \ {0} × (0, −∞), and a weak razor blade. (1) If p > P2 , all the solutions have a finite number of zeros. There exist positive integrable razor blades, with a singularity in |x|−δ . (2) If − p 0 < α, there is no regular solution on R N ×(0, −∞). There exist positive integrable razor blades as above. If α > 0, there exist positive solutions u with a weak singularity in |x|−δ , integrable if and only if α > N , and limt→0 |x|α u = L. If − p 0 < α < 0, there exist solutions with one zero and the same behavior. If α = − p 0 , all regular solutions, given by (1–8), have one zero. (3) If p < P2 , there exists a critical value α crit such that α ∗ < α crit < − p 0 for which the regular solutions u αcrit have constant sign, are integrable, and vanish identically at t = 0, with ku( . , t)k L ∞ (R N ) ≤ C|t|α/|β| . (4) If p < P2 and α ∈ (α ∗ , α crit ), there exist positive solutions u such that r δ w is periodic in ln r , and thus C 1 U∞ ≤ u ≤ C 2 U∞

for someC1 , C2 > 0.

There exist positive solutions with a weak singularity in |x|−δ , with the same bounds, such that r δ w is asymptotically periodic near ∞. The regular solutions have constant sign, are not integrable, vanish identically at t = 0, and r δ w is asymptotically periodic near ∞. (5) If p < P2 and α < α crit , there exist solutions oscillating around 0, such that r δ w is periodic. There exist solutions oscillating around 0, integrable, such that r δ w is asymptotically periodic. If α ≤ α ∗ , all regular solutions have constant sign, are not integrable, and vanish identically at t = 0.


p -LAPLACE EQUATION

209

Note. If p < P1 , recall that the Harnack inequality does not hold, as can be shown by the regular positive solutions constructed in Theorem 1.6, in particular those given by (1–6) when α = N . The two kinds of regular, integrable, solutions constructed for the critical values αcrit > 0 and α crit < 0 are of different types: the first, constructed for p > P2 , disappears in a spreading way, the second, for p < P2 , disappears in a focusing way. The case p > 2 will be treated in a second article [Bidaut-Véron 2006b], where we complete the results of [Gil and Vázquez 1997]. 2. General properties Different formulations of the problem. In the remainder of the article we can assume that α 6= 0, since the solutions are given explicitly by (1–9) when α = 0. Defining (2–1)

JN (r ) = r N (w + εr −1 |w0 | p−2 w0 ), Jα (r ) = r α−N JN (r ),

(E w ) can be written in an equivalent way under the form (2–2)

JN0 (r ) = r N −1 (N − α)w, Jα0 (r ) = −ε(N − α)r α−2 |w 0 | p−2 w 0 .

If α = N , then JN is constant, so we find again (1–5). We shall often use the following logarithmic substitution; for given d ∈ R, setting (2–3)

w(r ) = r −d yd (τ ),

Yd = −r (d+1)( p−1) |w0 | p−2 w0 ,

τ = ln r,

we obtain the equivalent system (2–4)

yd0 = dyd − |Yd |(2− p)/( p−1) Yd , Yd0 = ( p − 1)(d − η)Yd + εe( p+( p−2)d)τ (αyd − |Yd |(2− p)/( p−1) Yd ).

And yd , Yd satisfy the equations (2–5) yd00 + (η − 2d)yd0 − d(η − d)yd ε (( p−2)d+ p)τ + e |dyd − yd0 |2− p (yd0 + (α − d)yd ) = 0, p−1 2− p Yd0 (2–6) Yd00 + ( p − 1)(η − 2d − p 0 )Yd0 + εe(( p−2)d+ p)τ |Yd | p−1 + (α − d)Yd p−1 −( p − 1)2 (η − d)( p 0 + d)Yd = 0.

210


Reduction to an autonomous system. The substitution (2–3) with d = δ is the most helpful: setting (2–7)

y = yd ,

w(r ) = r −δ y(τ ),

Y = −r (δ+1)( p−1) |w0 | p−2 w0 ,

τ = ln r,

we are led to the autonomous system that plays a key role in the sequel: (S)

y 0 = δy − |Y |(2− p)/( p−1) Y, Y 0 = (δ − N )Y + ε(αy − |Y |(2− p)/( p−1) Y ).

Since N − δp = η − 2δ and N − δ = ( p − 1)(η − δ), Equation (2–5) takes the form (E y ) ( p − 1)y 00 + (N − δp)y 0 + δ(δ − N )y + ε|δy − y 0 |2− p (y 0 + (α − δ)y) = 0, while Equation (2–6) becomes (E Y )

Y 00 + (N − 2δ)Y 0 +

ε |Y |(2− p)/( p−1) Y 0 p−1 +ε(α − δ)|Y |(2− p)/( p−1) Y + δ(δ − N )Y = 0.

When w has constant sign, we define two functions associated to (y, Y ): |Y |(2− p)/( p−1) Y r w0 (r ) (τ ) = − , y w(r ) Y |w 0 (r )| p−2 w0 (r ) σ (τ ) = (τ ) = − . y r w(r )

ζ (τ ) = (2–8)

They play an essential role in the asymptotic behavior: ζ describes the behavior of w 0 /w and σ is the slope in the phase plane (y, Y ). They satisfy the equations ε ε(α − ζ ) 0 2− p ζ = ζ (ζ − η) + (2–9) |ζ y| (α − ζ ) = ζ ζ − η + , p−1 ( p − 1)σ 2− p

(2–10)

σ 0 = ε(α− N )+(|σ y| p−1 σ − N )(σ −ε) = ε(α− N )+(ζ − N )(σ −ε).

Note. Since (S) is autonomous, for any solution w of (E w ) of the problem, all the functions wξ (r ) = ξ δ w(ξr ), ξ > 0, are also solutions. From uniqueness, all regular solutions are completely described from one of them: w(r, a) = aw(a 1/δ r, 1); thus they present the same behavior at infinity. System (S) will be studied by using phase plane techniques, which was not done in [Qi and Wang 1999], and gives our main results. The set of trajectories of system (S) in the phase plane (y, Y ) is symmetric with respect to (0, 0). We define (2–11)

M = {(y, Y ) ∈ R2 : |Y |(2− p)/( p−1) Y = δy},


p -LAPLACE EQUATION

211

which is the set of the extremal points of y. We denote the four quadrants by Q1 = (0, ∞) × (0, ∞),

Q2 = (−∞, 0) × (0, ∞),

Q3 = −Q1 ,

Q4 = −Q2 .

Remarks 2.1. (i) The field at any point (ξ, 0) with ξ > 0 satisfies y 0 = −ξ 1/( p−1) < 0, and so points toward Q2 . The field at any point (ϕ, 0) with ϕ > 0 satisfies Y 0 = εαϕ, and so points toward Q1 if εα > 0 and toward Q4 if εα < 0. (ii) The pair (y, Y ) defined by (2–7) is related to JN by the identity (2–12)

JN (r ) = r N −δ (y(τ ) − εY (τ )),

τ = ln r,

and the formulae (2–2) can be recovered from the relations (2–13)

(y −εY )0 = (δ −α)y + ε(N −δ)Y = (δ −α)(y −εY ) + ε(N −α)Y = (δ − N )(y −εY ) + (N −α)y.

(iii) In the sequel the monotonicity of the functions yd , Yd , in particular y, Y , ζ and σ plays an important role. At any extremal point τ , these functions satisfy ε(α−d) (( p−2)d+ p)τ yd00 (τ ) = yd (τ ) d(η − d) − (2–14) e |dyd(τ ) |2− p , p−1 (2–15)

Yd00 (τ ) = Yd (τ ) ( p − 1)2 (η − d)( p 0 + d)

−ε(α − d)e(( p−2)d+ p)τ |Yd (τ )|(2− p)/( p−1) , (2–16) ( p−1)y 00 (τ ) = δ 2− p y(τ ) δ p−1 (N − δ) − ε(α − δ)|y(τ )|2− p = −|Y (τ )|(2− p)/( p−1) Y 0 (τ ), Y 00 (τ ) = Y (τ ) δ(N − δ) − ε(α − δ)|Y (τ )|(2− p)/( p−1) = εαy 0 (τ ), (2–18) ( p−1)ζ 00 (τ ) = ε(2− p) (α − ζ )|ζ |2− p |y|− p yy 0 (τ ) = ε(2− p) (α − ζ )(δ − ζ )|ζ y|2− p (τ ), (2–19) ( p−1)σ 00 (τ ) = (2− p) (σ − ε)|σ |(2− p)/( p−1) Y |y|(4−3 p)/( p−1) y 0 (τ )

(2–17)

= ζ 0 (τ )(σ (τ ) − ε). Energy functions for the system (S). There is a classical energy function associated to equation (E w ): (2–20)

E(r ) =

1 0 p α |w | + ε w2 , 0 p 2

which is nonincreasing when ε = 1, since E 0 (r ) = −(N − 1)r −1 |w 0 | p − εr w02 . This is not sufficient in our study: we need energy functions adapted to y and Y .

212


Using the ideas of [Bidaut-Véron 1989], we construct two of them by using the Anderson and Leighton formula [1968]. We find a first function W given by (2–21)

W (τ ) = W(y(τ ), Y (τ )), where 0 (2δ − N )δ p−1 p |Y | p α−δ 2 W(y, Y ) = ε |y| + 0 − δyY + y . p p 2

It satisfies W 0 (τ ) = ε(2δ− N )(δy −|Y |(2− p)/( p−1) Y )(|δy| p−2 δy −Y )−(δy −|Y |(2− p)/( p−1) Y )2 = (δy − |Y |(2− p)/( p−1) Y )(|δy| p−2 δy − Y ) δy − |Y |(2− p)/( p−1) Y × ε(2δ − N ) − . |δy| p−2 δy − Y When ε(2δ − N ) ≤ 0, then W is nonincreasing. When ε(2δ − N ) > 0, we consider the curve L = (y, Y ) ∈ R2 : H (y, Y ) = ε(2δ − N ) , where H (y, Y ) :=

δy − |Y |(2− p)/( p−1) Y |δy| p−2 δy − Y

and by convention this quotient takes the value |δy|2− p/( p − 1) if |δy| p−2 δy = Y . L is a closed curve surrounding (0, 0), symmetric with respect to (0, 0). Let SL be the domain with boundary L and containing (0, 0): (2–22) SL = (y, Y ) ∈ R2 : H (y, Y ) ≤ ε(2δ − N ) . Then W 0 (τ ) ≥ 0 if (y(τ ), Y (τ )) ∈ SL and W 0 (τ ) ≤ 0 if (y(τ ), Y (τ )) 6∈ SL . Observe that SL is bounded: indeed, for any (y, Y ) ∈ R2 , (2–23) H (y, Y ) ≥ 12 (δy)2− p + |Y |(2− p)/( p−1) . Also SL is connected; more precisely, for any (y, Y ) ∈ SL and any θ ∈ [0, 1], we have (θ y, θ p−1 Y ) ∈ SL . A second function, denoted by V , is also given by Anderson formula, or by multiplication by Y 0 in (E Y ): let (2–24)

V (τ ) = V(Y (τ ), Y 0 (τ )), where α−δ ε 0 V(Y, Z ) = δ(δ − N )Y 2 + Y 02 + |Y | p ; 0 2 p

then V 0 (τ ) = ε(2δ − N ) −

1 |Y |(2− p)/( p−1) Y 02 . p−1


p -LAPLACE EQUATION

213

When ε(2δ − N ) is not positive, V is nonincreasing. When it is positive, we have V 0 (τ ) ≥ 0 whenever |Y (τ )| ≤ D, where ( p−1)/(2− p) (2–25) D = ε(2δ − N )( p − 1) . The function W gives more information on the system, because SL is bounded, whereas the set of zeros of V 0 is unbounded. Stationary points of system (S). If α = δ = N , system (S) has infinitely many stationary points, given by ±(k, (δk) p−1 ), k ≥ 0. Otherwise, if ε(δ− N )(δ−α) ≤ 0, the system has a unique stationary point (0, 0). If ε(δ − N )(δ − α) > 0, it admits the three stationary points (2–26)

(0, 0),

M` = (`, (δ`) p−1 ) ∈ Q1 ,

M`0 = −M` ∈ Q3 ,

where ` is defined in (1–4). In that case, we find again that w ≡ `r −δ is a particular solution of equation (E w ). Local behavior at (0, 0). The linearized problem at (0, 0) is given by y 0 = δy,

Y 0 = (δ − N )Y + εαy,

and has eigenvalues µ1 = δ−N and µ2 = δ. Thus (0, 0) is a saddle point when δ < N and a source when N < δ. One can choose a basis of eigenvectors v1 = (0, −1) and v2 = (N , εα). Local behavior at M` . Setting y = ` + y,

(2–27)

Y = (δ`) p−1 + Y ,

system (S) is equivalent in Q1 to (2–28)

y 0 = δ y − εν(α)Y − 9(Y ),

Y 0 = εα y + (δ − N − ν(α))Y − ε9(Y ),

where δ(N − δ) , ( p − 1)(α − δ) 1/( p−1) (δ`)2− p 9(ϑ) = (δ`) p−1 + ϑ − δ` − ϑ, p−1 ν(α) =

(2–29)

with ϑ > −(δ`) p−1 . The linearized problem is given by y 0 = δ y − εν(α)Y ,

Y 0 = εα y + (δ − N − ν(α))Y .

Its eigenvalues λ1 ≤ λ2 are the solutions of the equation (2–30)

λ2 − (2δ − N − ν(α))λ + p 0 (N − δ) = 0.

214


The discriminant 1 of this equation is (2–31)

1 = (2δ − N − ν(α))2 − 4 p 0 (N − δ) = (N + ν(α))2 − 4ν(α)α.

The critical value α ∗ of α, given in (1–11), arises when ε(δ − N /2) > 0: α = α ∗ ⇐⇒ λ1 + λ2 = 0. When δ < N and ε = 1, then δ < α and M` is a sink when δ ≤ N /2 or δ > N /2 and α < α ∗ , and a source when δ > N /2 and α > α ∗ . When δ < N , and ε = −1, then α < δ and M` is a source when δ ≥ N /2 or δ < N /2 and α > α ∗ , and a sink when δ < N /2 and α < α ∗ . When N < δ, then M` is always a saddle point, but, as we will see later, the value α ∗ also plays a role. More specifically, the sign of α ∗ and its position with respect to N or η play a role. By computation, (2–32)

p 0 (δ 2 − 3δ + 2N ) (δ − N )2 =η+ 2(2δ − N ) ( p − 1)(2δ − N ) 2 (δ − N )(δ − (N + 3)δ + N ) . =N+ (2δ − N )(δ − 1)

α∗ =

Thus, if ε = 1, then α ∗ > η > 0 if N ≥ 2; if N = 1, α ∗ > 0 if p > 43 . If ε = −1, then α ∗ < − p 0 < 0. Otherwise, when 1 > 0 a basis of eigenvectors u 1 = (−εν(α), λ1 − δ), u 2 = (εν(α), δ − λ2 ) can be chosen. If 1 ≥ 0, then δ is exterior to the roots if εα > 0, and λ1 < δ < λ2 if εα < 0. Existence of solutions of equation (Ew ). Theorem 2.2. (i) Take r1 > 0 (r1 ≥ 0 if N = 1) and let a, a 0 be reals. There exists a unique solution w of equation (E w ) in a neighborhood V of r1 , such that w ∈ C 2 (V) and w(r1 ) = a, w0 (r1 ) = a 0 . It has a unique extension to a maximal interval of the form (Rw , ∞) with 0 ≤ Rw

if ε = 1,

(0, Sw )

if ε = −1.

with Sw ≤ ∞

If 0 < Rw or Sw < ∞, as the case may be, w is monotone near Rw or Sw with an infinite limit. (ii) For any a ∈ R, there exists a unique regular solution w of (E w ) satisfying (1–10), and (2–33)

lim |w 0 | p−2 w 0 /r w = −εα/N .

r →0


p -LAPLACE EQUATION

215

(iii) If N ≥ 2, any solution defined near 0 and bounded is regular. If N = 1, it satisfies lim r →0 w0 = b ∈ R, and lim r →0 w = a ∈ R. Proof. (i) Local existence and uniqueness near r1 > 0 follow directly from Cauchy’s theorem applied to equation (E w ) or to system (S), since the map ξ 7→ f p (ξ ) = |ξ |(2− p)/( p−1) ξ is of class C 1 . If N = 1, we can take r1 = 0, obtain a local solution in a neighborhood of 0 in R and reduce it to [0, ∞). Any local solution around r1 has a unique extension to a maximal interval (Rw , Sw ). Suppose that 0 < Rw (or Sw < ∞) and that w is oscillating around 0 near Rw (or Sw ). Making the substitution (2–3), with d 6= 0, if τ is a maximal point of |yd |, we see that (2–14) holds. If we take d such that ε(d − α) > 0, the sequence (yd (τ )) stays bounded, since the exponential has a positive limit; for that reason yd stays bounded, w is bounded near Rw (or Sw ) and then also JN0 , JN and w0 , which is contradictory. Thus w keeps a constant sign, for example w > 0, near Rw (or Sw ). At each extremal point r such that w(r ) > 0, we find (|w 0 | p−2 w0 )0 (r ) = −εαw(r ); thus r is unique since α 6= 0. Thus w is strictly monotone near Rw (or Sw ), and w and |w 0 | tend to ∞. First suppose ε = 1. We show that Sw = ∞. This is easy when α > 0: since E is nondecreasing, w and w0 are bounded for r > r1 . Assume α < 0 and Sw < ∞. Then for example w is positive near Sw , nondecreasing, and lim r →Sw w = ∞. Then Jα is nonincreasing and nonnegative near Sw ; hence again w and w0 are bounded, which is contradictory. Next suppose ε = −1. If Rw > 0, for example, w is positive and nonincreasing and lim r →Rw w = ∞. Then either α < N and JN is nonnegative and nondecreasing near Rw , and thus bounded, or α ≥ N and Jα is nonnegative and nondecreasing near Rw , and still bounded. In either case we reach a contradiction, then Rw = 0. (ii) By symmetry we can suppose a ≥ 0. Let ρ > 0. By (2–1) and (2–2), any regular solution w on [0, ρ] satisfies Z r w(r ) = a − ε f p (sT (w)) ds, 0 (2–34) Z 1 T (w)(r ) = w(r ) + (α − N ) θ N −1 w(r θ) dθ. 0

Conversely, any function w ∈ C 0 ([0, ρ]) that solves (2–34) satisfies w ∈ C 1 ((0, ρ]) and |w0 | p−2 w0 (r ) = r T (w); hence |w 0 | p−2 w0 ∈ C 1 ((0, ρ]) and w satisfies (E w ) in (0, ρ]. And lim r →0 r T (w) = 0, thus w ∈ C 1 ([0, ρ]) and |w0 | p−2 w0 ∈ C 1 ([0, ρ]). Then w satisfies (E w ) in [0, ρ] and w0 (0) = 0. From (E w ), we have lim |w0 | p−2 w0 /r w = −εα/N ,

r →0

216

MARIE FRANÇOISE BIDAUT-VÉRON 0

0

and therefore w − a = O(r p ) near 0. We look for w of the form a + r p ζ (r ), with ζ ∈ Bρ,M = {ζ ∈ C 0 ([0, ρ]) : kζ kC 0 ([0,ρ]) = max |ζ (r )| ≤ M}. r ∈[0,ρ]

We are led to the problem ζ = 2(ζ ), where Z 1 0 2(ζ )(r ) = −ε θ 1/( p−1) f p T (a + (r θ) p ζ (r θ)) dθ 0 1

Z = −ε

θ 1/( p−1) f p

0

0 αa + T ((r θ) p ζ (r θ)) dθ. N

Taking for example M = (|α|a)1/( p−1) , it follows that 2 is a strict contraction from Bρ,M into itself for ρ small enough, hence existence and uniqueness hold in [0, ρ]. (iii) If w is defined in (0, ρ) and bounded, then JN0 is integrable. Set l = lim JN (r ). r →0

Then |w0 | p−2 w0 = εlr 1−N (1 + o(1). If N ≥ 2, this implies l = 0; thus from above, w is regular. If N = 1, then lim r →0 w0 = b ∈ R, and lim r →0 w = a ∈ R. Definition. Suppose p > 1. Let Tr be the trajectory in the plane (y, Y ) (see (2–7)) starting from (0, 0) at −∞, with slope εα/N and y > 0 near time −∞. Its opposite −Tr is also a trajectory with the same properties (except that y < 0). Both are called regular trajectories. In this situation we say that y is regular. Observe that Tr starts in Q1 if εα > 0, and in Q4 if εα < 0. Remark 2.3. Let w be any solution of (E w ) such that w > 0 on some interval I . (i) The function w has at most one extremal point on I , since (|w0 | p−2 w0 )0 = −εαw, and this point is a maximum if εα > 0 and a minimum if εα < 0. (ii) From (2–33), if w is regular and w > 0 in (0, r1 ), r1 ≤ ∞, then w 0 < 0 in (0, r1 ) when εα > 0; thus Tr is in Q1 . And w 0 > 0 in (0, r1 ) when εα < 0; hence Tr is in Q3 in (−∞, ln r1 ). Remark 2.4. In the case δ 6= N , we can give a shorter proof of Theorem 2.2(ii). Indeed, (0, 0) is either a source or a saddle point. Thus there exists precisely one trajectory starting from (0, 0) at −∞, with y > 0, with slope εα/N . The corresponding solutions are regular: the slope σ defined in (2–8) satisfies limτ →−∞ σ = εα/N . Thus lim r →0 |w0 | p−2 w0 /r w = −εα/N , implying that w(2− p)/( p−1) has a limit a > 0. Since lim r →0 w 0 = 0, this function w satisfies (1–10), and any a is obtained by scaling. Notation. For any point P0 = (y0 , Y0 ) ∈ R2 \ {(0, 0)}, the unique trajectory in the phase plane (y, Y ) going through P0 is denoted by T[P0 ] . Notice that T[−P0 ] = −T[P0 ] , from the symmetry of system (S).


p -LAPLACE EQUATION

217

First sign properties. Proposition 2.5. Let w 6≡ 0 be any solution of (E w ). (i) If ε = 1 and α ≤ max(N , η), then w has at most one zero, and no zero if w is regular. (ii) If ε = 1 and N < min(δ, α) and w is regular, then w has at least one zero. (iii) If ε = −1 and α ≥ min(0, η), then w has at most one zero. If α > 0 and w is regular, then it has no zero. (iv) If ε = −1 and − p 0 ≤ α < min(0, η), then w0 has at most one zero; consequently w has at most two zeros, and at most one if w is regular. Proof. (i) Let ε = 1. Take two consecutive zeros ρ0 < ρ1 of w, with w > 0 on (ρ0 , ρ1 ), so w 0 (ρ1 ) < 0 < w0 (ρ0 ). If α ≤ N , we find, using the function JN of (2–1), Z ρ1 N −1 N −1 0 0 p−2 p−1 JN (ρ1 )− JN (ρ0 ) = −ρ1 |w (ρ1 )| −ρ0 w (ρ0 ) = (N −α) s N −1 w ds, ρ0

which is contradictory; thus w has at most one zero. If w is regular with w(0) > 0 and ρ1 is a first zero, then Z ρ1 JN (ρ1 ) = −ρ1N −1 |w 0 (ρ1 )| p−1 = (N − α) s N −1 w ds ≥ 0, 0

again a contradiction. Next suppose 0 < α ≤ η and use the substitution (2–3), with d = α. Then yα has at most one zero: indeed, if yα has a maximal point τ where it is positive, and is not constant, then from (2–14), (2–35)

yα00 (τ ) = α(η − α)yd (τ );

hence yα00 (τ ) < 0, which is impossible. In the same way the regular solution satisfies limτ →−∞ yα = 0 since α > 0, and yα has no maximal point; thus yα is positive and increasing. Rr (ii) Let ε = 1 and w > 0 on [0, ∞). If N < α, then JN (r ) = (N −α) 0 s N −1 w ds < 0. 0 The function r 7→ δr p − w( p−2)/( p−1) is nonincreasing; hence w = O(r −δ ) at ∞, so y is bounded at ∞. For any r ≥ 1, one gets JN (r ) ≤ JN (1) < 0, hence y(τ ) + |JN (1)|e(δ−N )τ ≤ Y (τ ) for any τ ≥ 0, from (2–12). Then limτ →∞ Y = ∞, implying by (S) that limτ →∞ y 0 = −∞, which is impossible. (iii) Let ε = −1 and α ≥ min(η, 0). We use again the substitution (2–3) for some d 6= 0. If yd is not constant and has a maximal point where it is positive, then (2–14) holds. Taking d ∈ (0, min(α, η)) if N ≥ 2 and α > 0 and d = −1 if N = 1 and η = −1 ≤ α, we reach a contradiction. Now suppose w is regular and α > 0. Then w0 > 0 near 0, from Theorem 2.2, and as long as w stays positive, any extremal point r is a strict minimum; thus in fact w 0 > 0 on [0, Sw ).

218


(iv) Let ε = −1 and − p 0 ≤ α < min(0, η). Suppose that w 0 has two consecutive zeros ρ1 < ρ2 , and use (2–3) again with d = α. If the function Yα is not constant and has a maximal point τ where it is positive, we get from (2–15) (2–36)

Yα00 (τ ) = ( p − 1)2 (η − α)( p 0 + α)Yα (τ );

thus Yα00 (τ ) < 0, and Yα has at most one zero. Next consider regular solutions: they satisfy Yα = e(α( p−1)+ p)τ (|α|a/N )(1 + o(1)) near −∞, by Theorem 2.2 and (2–3). Thus limτ →−∞ Yα = 0; as above Yα cannot have any extremal point, so Yα is positive and increasing. Then w 0 < 0 from (2–3), and w has at most one zero. Remark 2.6. From (2–35) and (2–36) we see that if 0 < α ≤ η then yα has only minimal points on any set where it is positive, and the same conclusion holds for Yα when − p 0 ≤ α ≤ min(η, 0)). Proposition 2.7. Let y be any solution of (E y ), linked with w by (2–7), and having constant sign in a semi-interval around the point ln Rw or ln Sw . (i) If y is not strictly monotone near that same point, then Rw = 0 or Sw = ∞. If y is not constant, then either ε = 1 and δ < N < α or ε = −1 and α < δ < N . In any case, y oscillates around `. (ii) If y is strictly monotone near ln Rw or ln Sw , then also Y, ζ, σ are monotone near the same point. Proof. Let s = Rw or Sw , and suppose that y has constant sign near s. Then so does Y , by Remark 2.3. (i) At each point τ where y 0 (τ ) = 0, we have y 00 (τ ) 6= 0, and (2–16) holds with y > 0. Suppose that y is not strictly monotone near s. There exists a strictly monotone sequence (τn ) converging to s and such that y 0 (τn ) = 0, y 00 (τ2n ) > 0, y 00 (τ2n+1 ) < 0. Then either ε = 1 and δ < min(α, N ), or ε = −1 and α < δ < N ; and y(τ2n ) < ` < y(τ2n+1 ). This cannot happen if s is finite, because y tends to ∞. It is also impossible when ε = 1 and α ≤ N ; indeed, there exist at least two points θ1 < θ2 such that y(θ1 ) = y(θ2 ) = ` and y ≥ ` on (θ1 , θ2 ), with y 0 (θ1 ) > 0 > y 0 (θ2 ). Then from (S), Y (θ1 ) < (δ`) p−1 < Y (θ2 ). And from (2–13), (e(N −δ)τ (y − Y ))0 = (N − α)e(N −δ)τ y; and the constant (`, (δ`) p−1 ) is also a solution of (S), hence (e(N −δ)τ (y − ` − Y + (δ`) p−1 ))0 = (N − α)e(N −δ)τ (y − `) ≥ 0 on (θ1 , θ2 ). A contradiction follows by integration on this interval. (ii) Suppose y strictly monotone near s. At any extremal point τ of Y , we find Y 00 (τ ) = εαy 0 (τ ) from (2–17); hence y 0 (τ ) 6= 0, and Y 00 (τ ) has constant sign; thus τ is unique, and Y is strictly monotone near s.


p -LAPLACE EQUATION

219

Next consider the function ζ satisfying (2–9). If there exists τ0 such that ζ (τ0 ) = α, then ζ 0 (τ0 ) = α(α − η). If α 6= η, then τ0 is unique, so α − ζ has a constant sign near s. Then also ζ 00 (τ ) has constant sign at any extremal point τ of ζ , from (2–18). Then ζ is strictly monotone near s. If α = η, then ζ ≡ α. Finally consider σ , which satisfies (2–10). At each point τ such that σ 0 (τ ) = 0, (2–19) holds and Y has a constant sign. If there exists τ0 such that σ (τ0 ) = ε, then σ 0 (τ0 ) = ε(α − N ). If α 6= N , then τ0 is unique, and σ − ε has constant sign near s. Thus σ 00 (τ ) has constant sign at any extremal point τ of σ , by (2–19), since Y has constant sign near s. If α = N , then σ ≡ ε. Behavior of w near 0 or ∞. Here we suppose w is defined near 0 or ∞, which means the function y of (2–7) is defined near ±∞. We study the behavior of y and then return to w. First we suppose y monotone, so we can assume y > 0 near ±∞. We do not look for a priori estimates, which could be obtained by successive approximations as in [Bidaut-Véron 2006a]. Our method is based on monotonicity and L’Hospital’s rule, which is much more rapid and efficient. Proposition 2.8. Let (y, Y ) be any solution of (S) such that y is strictly monotone and y > 0 near s = ±∞. Then ζ has a finite limit λ near s, which is equal to 0, α, η, δ. More precisely, we are in one of the following cases: (i) (y, Y ) converges to a stationary point different from (0, 0). Then λ = δ, and ε(δ − N )(δ − α) > 0 or α = δ = N . (ii) (y, Y ) converges to (0, 0). Then • either λ = 0, s = −∞, and y is regular, or N = 1; • or λ = η; then either (s = ∞, δ < N ) or (s = ∞, δ = N , ε(α − N ) < 0)) or (s = −∞, N < δ) or (s = −∞, δ = N , ε(α − N ) > 0)). (iii) limτ →s y = ∞ and λ = α. Then either (s = ∞, α < δ) or (s = ∞, α = δ, ε(δ − N ) < 0) or (s = −∞, δ < α) or (s = −∞, α = δ, ε(δ − N ) > 0). Proof. From Proposition 2.7, the functions y, Y, σ, ζ are monotone; hence ζ has a limit λ ∈ [−∞, ∞] and σ has a limit µ ∈ [−∞, ∞], and (y, Y ) converges to a stationary point, or lim y = ∞. Then lim |Y | = ∞, since α 6= 0 from system (S). To apply the L’Hospital’s rule, we consider the two quotients (2–37)

Y 0 (δ − N )σ + ε(α − ζ ) = y0 δ−ζ

and (2–38)

(|Y |(2− p)/( p−1) Y )0 ζ (δ − N + ε(α − ζ )/σ ) = y0 ( p − 1)(δ − ζ ) ζ (δ − N ) + ε(α − ζ )|ζ y|2− p = . ( p − 1)(δ − ζ )

220


(i) First case: ε(δ − N )(δ − α) > 0 and (y, Y ) converges to the point M` defined by (2–26). Then obviously λ = δ; or α = δ = N and limτ →s y = k > 0; then limτ →s Y = (δk) p−1 , so λ = δ. (ii) Second case: (y, Y ) converges to (0, 0). Then λ is finite; indeed, if λ = ±∞, the quotient (2–38) converges to (N − δ)/( p − 1), because |ζ y| = |Y |1/( p−1) = o(1); thus ζ = |Y |(2− p)/( p−1) Y/y has the same limit, from L’Hospital’s rule, which is contradictory. We next consider N in relation to δ. If N < δ, then (0, 0) is a source, thus s = −∞. Using the eigenvectors, either µ = εα/N , then |ζ | p−1 = |µ|y 2− p (1 + o(1)), thus λ = 0 and w is regular, from Remark 2.4. Or µ = ±∞; then λ = λ(δ − N )/( p − 1)(δ − ζ ) from (2–38), thus λ = 0 or λ = η. If λ = 0, then ζ 0 /ζ converges to −η from (2–9), and s = −∞, thus necessarily η < 0, which means N = 1. If δ < N (so N ≥ 2), then (0, 0) is a saddle point. Thus either s = −∞ and µ = εα/N , λ = 0 and w is regular. Or s = ∞, µ = ±∞, and as above, λ = 0 or λ = η. Now if λ = 0 the quotient (2–37) converges to ∓∞, which is contradictory. Thus λ = η. If δ = N (so N ≥ 2), either λ = 0, so y 0 > 0, s = −∞, and µ = εα/N by (2–38); or else λ > 0, in which case λ = N = η from (2–38). Moreover if s = ∞, then ε(α − N ) < 0; if s = −∞, then ε(α − N ) > 0. Indeed (εy − Y )0 = ε(N − α)y and y − εY converges to 0; thus if s = ∞ and ε(N − α) ≥ 0, or s = −∞ and ε(N − α) ≤ 0, then µ ≤ ε, but µ = ∞, we reach again a contradiction. (iii) Third case: y tends to ∞. If s = ∞, then y 0 > 0, thus ζ < δ; if s = −∞, then ζ > δ. If λ = ±∞, then the quotient (2–38) converges to ε∞; thus λ = ε∞ and s = −ε∞. In any case, ζ 0 < 0, so |µ| ≤ 1/( p − 1) by (2–9), and µ = ε by (2–37); thus Y 0 = −ε|Y |(2− p)/( p−1) Y (1 + o(1)), and we reach a contradiction by integration. Thus λ is finite; moreover λ 6= 0 for otherwise we would have µ = 0, seeing that σ = |ζ y| p−2 ζ ; but µ = α/δ by (2–37). If α 6= δ, then λ = α or δ, by (2–38). In turn σ = |λy| p−2 λ(1+o(1)), thus µ = 0. From (2–37), necessarily λ = α. And if s = ∞, then y 0 > 0, thus ζ < δ, thus α < δ. If s = −∞, then similarly α > δ. If α = δ, then λ = α = δ 6= N , and ε(δ − N )(δ − ζ ) < 0 from (2–38); thus if s = ∞, then ε(δ − N ) < 0 since ζ < δ; if s = −∞, then ε(δ − N ) > 0. Next we improve Proposition 2.8 by giving a precise behavior of w in any case: Proposition 2.9. We keep the assumptions of Proposition 2.8. (i) If λ = α 6= δ, then lim r α w = L > 0 (near 0, or ∞). (ii) If λ = η > 0, η 6= N , then lim r η w = c > 0.


p -LAPLACE EQUATION

221

(iii) If λ = α = δ 6= N , then (2–39)

lim r δ (ln r )−1/(2− p) w = κ := (2 − p)δ p−1 |N − δ|

1/(2− p)

.

(iv) If λ = η = N = δ 6= α, then (2–40)

(N +1)/2

lim r (ln r ) N

1 w = ρ := N

N (N − 1) 2|α − N |

(N +1)/2

.

(v) If N = 1, λ = η = −1 or λ = 0 (near 0) then (2–41)

lim w = a ∈ R,

r →0

lim w 0 = b;

r →0

and b 6= 0; moreover, a = 0 (hence b > 0) if and only if λ = −1. Proof. (i) Let λ = α 6= δ. From (2–8) we have r w 0 (r ) = −αw(r )(1 + O(1)). Next we apply Proposition 2.8, and are led to two cases: If s = ∞ and α < δ, then for any γ > 0 we have w = O(r −α+γ ) and 1/w = O(r α+γ ) near ∞ and w0 = O(r −α−1+γ ). Then Jα0 (r ) = O(r α(2− p)− p−1+γ ), so Jα0 is integrable, hence Jα has a limit L, and lim r α w = L, seeing that Jα (r ) = r α w(1 + o(1)). If L = 0, then r α w = O(r α(2− p)− p+γ ), which contradicts the estimate of 1/w = O(r α+γ ) for γ small enough. Thus L > 0. Otherwise, we have s = −∞ and δ < α; hence limτ →s y = ∞, w = O(r −α−γ ), 1/w = O(r α−ν ), w0 = O(r −α−1−γ ) near 0, and Jα0 (r ) = O(r α(2− p)− p−1−γ ). Thus Jα0 is still integrable; hence lim r α w = L ≥ 0. If L = 0, then r α w = O(r α(2− p)− p−γ ), which contradicts the estimate of 1/w. Therefore we again obtain L > 0. (ii) Let λ = η > 0, η 6= N . From Proposition 2.8, either s = ∞, δ < N or s = −∞, N < δ. As above we get w = O(r −η±γ ) and 1/w = O(r η±γ ) near ∞ or 0. Here we make the substitution (2–3) with d = η. We find yη = O(e±γ τ ), 1/yη = O(e±γ τ ), yη0 = O(e±γ τ ), thus Yη = O(e±γ τ ), and from (2–4), Yη0 = O(e±γ τ ). Substituting in (2–4), we deduce Yη0 = O(e(2− p)((δ−η)±γ )τ ). When s = ∞, then δ < η, when s = −∞, then δ > η from (1–2). In any case, Yη0 is integrable, hence Yη has a limit k, and Yη −k = O(e(2− p)((δ−η)±γ )τ ). Now (e−ητ yη )0 = −e−ητ Yη , thus yη has a limit c = k 1/( p−1) /η; in other words, lim r η w = c. If c = 0, then Yη = O(e(2− p)((δ−η)±γ )τ ), yη = O(e((2− p)((δ−η)±γ )/( p−1))τ ), which contradicts 1/yη = O(eγ τ ) for γ small. 1/( p−1)

(iii) Now suppose λ = α = δ 6= N . Then either s = ∞ and ε(δ − N ) < 0 or s = −∞ and ε(δ − N ) > 0; moreover, limτ →s y = ∞. Then Y = (δy) p−1 (1 + o(1)), and µ = 0; hence y − εY = y(1 + o(1)), and from (2–13), (y − εY )0 = ε(N − δ)Y = ε(N − δ)δ p−1 (y − εY ) p−1 (1 + o(1)). Then y = (|N − δ|δ p−1 (2 − p)|τ |)1/(2− p) (1 + o(1)), which is equivalent to (2–39) by (2–7).

222


(iv) Let λ = η = N = δ 6= α. Then either s = ∞ and ε(α − N ) < 0 or s = −∞ and ε(α − N ) > 0; moreover, limτ →s y = 0. Then Y = (N y) p−1 (1 + o(1)) and µ = ∞, so Y − εy = Y (1 + o(1)), and from (2–13) we have (Y − εy)0 = ε(α − N )y = ε(α − N )N −1 (Y − εy)1/( p−1) (1 + o(1)). (N +1)/2 Hence y = c|τ |−(N +1)/2 (1 + o(1)) with c = N −1 N (N −1)/2|α−N | , and (2–40) follows from (2–7). (v) Let λ = 0. Then also r w 0 = o(w); thus by integration we get w +|w0 | = O(r −k ) for any k > 0. Then J10 is integrable, so J1 has a limit at 0, and lim r →0 r w = 0. Therefore lim r →0 w 0 = b ∈ R and lim r →0 w = a ≥ 0. Then b 6= 0, since regular solutions satisfy (2–33), and a 6= 0, since a = 0 would imply w = −br (1 + o(1)), ζ = −1. If λ = η = −1, then from (2–8), w is nondecreasing, so it has a limit a ≥ 0 at 0, leading to w0 = −aλr −1 (1 + o(1)), and by integration a = 0. And ((w0 ) p−1 )0 = ε(1 − α)w(1 + o(1)), so w0 has a limit b 6= 0. Next we consider the cases where y is not monotone and possibly changes sign. Proposition 2.10. Assume ε = 1. (i) Suppose that N ≤ δ < α, or N < δ ≤ α. Then any solution y has a infinite number of zeros near ∞. (ii) Suppose that y has a infinite number of zeros near ±∞. Then either N < α < δ and |y| < ` and |Y | < (δ`) p−1 near ± ∞, or N < δ = α, or max(δ, N , η) < α. If moreover δ < N < α, then |y| exceeds ` at its extremal points and |Y | exceeds (δ`) p−1 at its extremal points. Proof. (i) Suppose the conclusion does not hold. Then for example y > 0 for large τ ; and y is monotone, from Proposition 2.7(i). Applying Proposition 2.8 with s = ∞, we reach a contradiction. (ii) Suppose that y is oscillating around 0 near ±∞. Then from (2–16), at the extremal points, (2–42)

|y(τ )|2− p (δ − α) < (δ − N )δ p−1 ,

and the inequality is strict, because in case of equality, y is constant by uniqueness. Similarly Y is oscillating around 0, and at the extremal points one finds, from (2–17), (2–43)

|Y (τ )|(2− p)/( p−1) (δ − α) < (δ − N )δ.

Then max(N , η) < α, thanks to Proposition 2.5; and the conclusions follow from (2–42) and (2–43).


p -LAPLACE EQUATION

223

We can complete these results according to the sign of δ − N /2: Proposition 2.11. Suppose that ε(δ − N /2) ≤ 0. Then any solution y has a finite number of zeros near ln Rw or ln Sw . If y is defined near ±∞ and nonmonotone, then (y, Y ) converges to ±M` . There is no cycle or homoclinic orbit in R2 . Proof. (i) Suppose that y has an infinity of zeros. Then Rw = 0 or Sw = ∞, and there exists a strictly monotone sequence (rn ) of consecutive zeros of w, converging to 0 or ∞. Since ε(δ − N /2) ≤ 0, the energy function V defined in (2–24) is nonincreasing. We claim that V is bounded. This is not easy to prove; we define for the purpose a function U (r ) = r N 12 w 2 + εr −1 |w0 | p−2 w0 w = e(N −2δ)τ y 21 y − εY . Then 0 U 0 (r ) = r N −1 ( 12 N − α)w 2 + ε|w 0 | p = e(N −1−2δ)τ ( 12 N − α)y 2 + ε|Y | p . If ε = 1, then δ ≤ N /2 < N < α. If ε = −1, then α < 0, by Proposition 2.10. Then U (rn ) = 0 and εU 0 (rn ) > 0. Therefore there exists another sequence (sn ) such that sn ∈ (rn , rn+1 ), U (sn ) = 0, and εU 0 (sn ) ≤ 0. At the point τn = esn we find 0 0 0 21− p y 2 p = 2|Y | p ≤ ε(2α − N )y 2 , so (y(τn ), Y (τn )) is bounded. Hence (V (τn )) is bounded, so V is bounded near ±∞. Therefore V has a finite limit χ , and Y and Y 0 are bounded because ε(α − δ) > 0; in turn, (y, Y ) is bounded. Otherwise (0, 0) and ±M` are not in the limit set at ±∞, since (0, 0) is a saddle point, and ±M` is a source or a sink. Then the trajectory has a limit cycle, and there exists a periodic solution (y, Y ). The corresponding function V is periodic and monotone, hence constant; then V 0 ≡ 0 implies that Y is constant and hence also y, by (S). But this is a contradiction. (ii) Suppose that y is positive near ±∞, and nonmonotone. If ε = 1, then δ ≤ N /2 < N < α; if ε = −1, then α < δ < N , by Proposition 2.7, and y oscillates around `. There exists a sequence of minimal points (τn ), where y(τn ) < `, and |Y (τn )| = δy(τn ); thus again (y(τn ), Y (τn )) is bounded, and as above (y, Y ) is bounded. The trajectory has no limit cycle, and hence converges to M` . Finally, if there is an homoclinic orbit, then Tr is homoclinic. Then limτ →−∞ V = limτ →∞ V = 0; hence V ≡ 0, and as above (y, Y ) is constant, so (y, Y ) ≡ (0, 0), again a contradiction. Proposition 2.12. If y is not monotone near ε∞ (positive or changing sign), then y and Y are bounded. Proof. From Proposition 2.11, it follows that ε(δ − N /2) > 0. When ε = 1, and y is changing sign and N < α < δ, then |y| is bounded by ` from above. Apart from this case, if y is changing sign, then ε(α −δ) > 0, from Proposition 2.11. If y stays positive, either ε = 1, δ < min(α, N ), or ε = −1, α < δ < N , by Proposition 2.7.

224


In any case ε(α − δ) > 0. Here we use the energy function W defined by (2–21). We can write W(y, Y ) in the form (2–44)

W(y, Y ) = ε(F(y, Y ) + G(y)),

with 0

(2–45)

|Y | p |δy| p (δ−N )δ p−1 p ε(α−δ) 2 F(y, Y ) = 0 −δyY + , G(y) = |y| + y . p p p 2

Observe that F(y, Y ) ≥ 0, so εW(y, Y ) ≥ G(y) > 0 for large |y|. Then W 0 (τ ) ≤ 0 whenever (y(τ ), Y (τ )) 6∈ SL , where SL is given in (2–22). Let τ0 be arbitrary in the interval of definition of y. Since SL is bounded, there exists k > 0 large enough that εW (τ ) ≤ k for any τ such that ε(τ − τ0 ) ≥ 0 and (y(τ ), Y (τ )) ∈ SL , and we can choose k > W (τ0 ). Then εW (τ ) ≤ k for ε(τ − τ0 ) ≥ 0; hence y and Y are bounded near ε∞. Further sign properties. We can improve Proposition 2.5 using Propositions 2.8 and 2.9: Proposition 2.13. Assume ε = 1, −∞ < α ≤ δ and α < N . Then all regular solutions have constant sign, y is strictly monotone and limτ →∞ ζ = α. Any solution has at most one zero, and limτ →∞ ζ = α. Proof. Regular solutions have constant sign by Proposition 2.5. Moreover JN is increasing from 0; thus it is positive for r > 0, which means Y < y. And y is monotone near ∞, by Proposition 2.7. From Proposition 2.8, we have three possibilities: either α < N < δ and limτ →∞ ζ = δ, in which case limτ →∞ Y/y = (δ − α)/(δ − N ) > 1, which is impossible; or δ ≤ N and limτ →∞ ζ = η ≥ N , in which case limτ →∞ Y /y = ∞, which is also contradictory, or finally limτ →∞ ζ = α. Moreover y is increasing on R from 0 to ∞; indeed, if y has a local maximum for some τ , we get α < N < δ and y(τ ) ≤ ` from (2–16), and moreover ` < δ ( p−1)/(2− p) ; but δy(τ ) = Y (τ )1/( p−1) < y(τ )1/( p−1) , which is contradictory. For the second statement, we see from Proposition 2.5 that any solution w 6≡ 0 has at most one zero. If w(r1 ) = 0 and, say, w > 0 on (r1 , ∞), we get w0 (r1 ) > 0; thus JN (r ) ≥ JN (r1 ) > 0 for r ≥ r1 , and we conclude as above. Proposition 2.14. Assume ε = −1. (i) If α < 0 and N ≤ δ, all regular solutions have at least one zero. (ii) If 0 < α, all regular solutions have constant sign and satisfy Sw < ∞. (iii) If − p 0 < α < min(0, η), all regular solutions have precisely one zero and Sw < ∞.


p -LAPLACE EQUATION

225

Proof. (i) Let α < 0 and N ≤ δ. Since εα > 0, the trajectory Tr starts in Q1 . Suppose that y stays positive. Then Tr stays in Q1 , from Remark 2.3. If N ≤ δ, then y is monotone, since it can only have minimal points, from (2–16); and (0, 0) is the only stationary point. Then limτ →∞ y = ∞, and limτ →∞ ζ = α < 0 from Proposition 2.8; thus (y, Y ) is in Q4 for large τ , which is impossible. (ii) Let 0 < α. Then εα < 0, so that Tr starts in Q4 . Moreover y > 0 on R, by Proposition 2.5. And Tr stays in Q4 , by Remark 2.1(i) on page 211. Thus y 0 = δy+|Y |1/( p−1) > 0. If Sw = ∞, we see from Proposition 2.8 that limτ →∞ ζ = α > 0; hence (y, Y ) ends up in Q1 , which is false. Then Sw < ∞. (iii) Let − p 0 < α < min(0, η). Then Tr starts in Q1 . By Proposition 2.5, Yα stays positive, Tr stays in Q1 ∪ Q2 , and Yα is increasing: Yα0 = −( p − 1)(η − α)Yα + e( p−(2− p)α)τ (Yα1/( p−1) − αyα ) > 0. Suppose that Sw = ∞. Then limτ →∞ Yα (τ ) ≥ C > 0, so r α+1 w0 (r ) ≤ −C 1/( p−1) for large r , and, by integration, r α w(r ) ≤ −C 1/( p−1) /2. In particular, we obtain from (2–3) that limτ →∞ y = −∞. From Propositions 2.7, 2.8, and 2.9, it follows that lim r →∞ r α w = L < 0; thus limτ →∞ Yα (τ ) = (αL) p−1 . And there exists a unique τ0 such that yα (τ0 ) = 0, by Remark 2.1(i). But (2–46)

Yα00 (τ )−( p−1)2 (η−α)(α+ p 0 )Yα Y 0 1 ( p−(2− p)α)τ 1/( p−1) = α e Yα − ( p−1)(η−2α− p 0 )Yα Yα p−1 Y 0 α ( p−(2− p)α)τ ≥ α e yα + (η−α)(2− p) + ( p−1)(α+ p 0 )Yα . Yα p−1

Thus Yα00 (τ ) > 0 for any τ ≥ τ0 , an impossibility. Then Sw < ∞, limτ →ln Sw Y/y = −1, and y has a zero. Behavior of w near Rw > 0 or Sw < ∞. Proposition 2.15. Let w be any solution of (E w ) with a reduced domain (so either ε = 1 and Rw > 0 or ε = −1 and Sw < ∞). Let s = Rw or Sw . Then (2–47)

lim |r − s|

r →s

( p−1)/(2− p) 1/(2− p)

s

w=±

p−1 p−1 2− p

2− p

and

lim σ = ε.

τ →ln s

Proof. From Proposition 2.5, we can suppose that εw is decreasing near s and lim r →s w = ∞; thus y > 0, εY > 0 near ln s, and limτ →ln s y = ∞. Also, σ is monotone near ln s, by Proposition 2.7; thus it has a limit µ such that εµ ∈ [0, ∞]. Suppose that µ = 0. Then Y = o(y) = o(y − εY ); from (2–13) we get (y − εY )0 = (δ − α)(y − εY ) + ε(N − α)Y = (δ − α + o(1))(y − εY ),

226


so y cannot blow up in finite time. In the same way, if µ = ∞, then y = o(εY ) = o(εY − y), and (y − εY )0 = (δ − N )(y − εY ) + (N − α)y = (δ − N + o(1))(y − εY ), again leading to a contradiction; thus εµ ∈ (0, ∞). Therefore limτ →ln Rw ζ = ε∞ and µ = ε from (2–37); then w0 w −1/( p−1) + (ε + o(1))r 1/( p−1) = 0, and (2–47) holds. More information on stationary points. The Hopf bifurcation point. When ε(δ − N /2) > 0, a Hopf bifurcation appears at the critical value α = α ∗ given by (1–11). Then some cycles do appear near α ∗ , by the Poincaré–Andronov–Hopf theorem; see [Hale and Koçak 1991, p. 344]. We get more precise results by using the Lyapunov test for a weak sink or source; it requires an expansion up to the order 3 near M` , in a suitable basis of eigenvectors, where the linearized problem has a rotation matrix. Theorem 2.16. Let ε(δ − N /2) > 0. (i) Suppose ε = 1. If α = α ∗ , then M` is a weak source. If α < α ∗ with α ∗ − α small enough, there exists a unique limit cycle in Q1 attracting at −∞. (ii) Suppose ε = −1. If α = α ∗ , then M` is a weak sink. If α > α ∗ with α − α ∗ small enough, there exists a unique limit cycle in Q1 , attracting at ∞. √ Proof. The eigenvalues are given by λ1 = −ib, λ2 = ib, with b = p 0 (N − δ). From (2–29) we get ν(α ∗ ) = 2δ − N =

δ(N − δ) ε(δ`)2− p = . ( p − 1)(α ∗ − δ) ( p − 1)

First we make the substitution (2–27) as above, which leads to (2–28). The function 9 defined in (2–29) has an expansion near t = 0 of the form 9(ϑ) = B2 ϑ 2 + B3 ϑ 3 + · · · , where B2 =

(2 − p)(δ`)3−2 p , 2( p − 1)2

B3 =

2(3 − 2 p)B22 (2 − p)(3 − 2 p)(δ`)4−3 p = . 6( p − 1)6 3(2 − p)ν(α ∗ )

Next we make the substitution τ = −θ/b,

y(τ ) = εν(α)x1 (θ),

Y (τ ) = δx1 (θ) + bx2 (θ),

and obtain x10 (θ ) = x2 +

ε 9(δx1 + bx2 ), bν(α)

x20 (θ) = −x1 −

ε(N − δ) 9(δx1 + bx2 ). b2 ν(α)


p -LAPLACE EQUATION

227

We write the expansion of order 3 in the form x10 = x2 + ε(a2,0 x12 +a1,1 x1 x2 +a0,2 x22 +a3,0 x13 +a2,1 x12 x2 +a1,2 x1 x22 +a0,3 x23 +· · · ), x10 = −x1 +ε(b2,0 x12 +b1,1 x1 x2 +b0,2 x22 +b3,0 x13 +b2,1 x12 x2 +b1,2 x1 x22 +b0,3 x23 +· · · ), and we compute the Lyapunov coefficient L C = ε(3a3,0 + a1,2 + b2,1 + 3b0,3 ) −a2,0 a1,1 + b1,1 b0,2 − 2a0,2 b0,2 − a0,2 a1,1 + 2a2,0 b2,0 + b1,1 b2,0 . After simplification, we obtain (2 − p)bν(α)2 L C = (N − 2δ)(1 − ε(3 − 2 p)) 2B22 (δ 2 + b2 ) ( 2(N − 2δ)( p − 1) < 0 if ε = 1, = . 2(N − 2δ)(2 − p) > 0 if ε = −1, The nature of M` follows from [Hubbard and West 1995, p. 292], taking into account that θ has opposite sign from τ . If ε = 1, M` is a week source, and there exists a small limit cycle attracting at −∞ for all α near α ∗ such that M` is a sink; this means that α < α ∗ . If ε = −1, M` is a weak sink and there exists a small limit cycle attracting at ∞ for all α near α ∗ such that M` is a source; this means α ∗ < α. Node points or spiral points. When the system (S) has three stationary points, and M` is a source or a sink (so δ < N ), it is interesting to know if M` is a node point. When α ∗ exists, it is a spiral point, by (2–30). √ If ε = 1, we see from (2–31) that M` is a node point when δ ≤ N /2− p 0 (N − δ) √ 0 √ 0 or δ > N /2 − p (N − δ) and α ≤ α1 , or δ > N /2 + p (N − δ) and α2 ≤ α, where α1 = δ + (2–48)

δ(N − δ) , √ ( p − 1)(2δ − N + 2 p 0 (N − δ))

δ(N − δ) . √ ( p − 1)(2δ − N − 2 p 0 (N − δ)) √ √ If ε = −1, M` is a node when δ ≥ N /2 + p 0 (N − δ), or δ < N /2 + p 0 (N − δ) √ 0 and α2 ≤ α, or δ < N /2 − p (N − δ) and α ≤ α1 . In any case α1 < α2 . α2 = δ +

Remarks 2.17. (i) Let ε = 1. One can verify that N ≤ α1 and that N = α1 if and only if N = δ/( p−1) = p 0 /(2− p). Also α1 < η if and only if δ 2 +(7− N )δ+ N > 0, which is true for N ≤ 14, but not always.

228


(ii) Let ε = −1. It is easy to see that α2 ≤ 0 and that α2 = 0 if and only if N (2 − p) = δ, or equivalently N = p/(2 − p)2 . Also α2 > − p 0 if and only if δ 2 + 7δ − 8N < 0, which is true for δ < N /2 < 9, but not always. Nonexistence of cycles. If the system (S) admits a cycle O in R2 , then O surrounds at least one stationary point. If it surrounds (0, 0), the corresponding solutions y are not of constant sign. If it only surrounds M` , then it stays in Q1 , so y stays positive. Indeed α 6= 0 from (1–9), and O cannot intersect {(ϕ, 0), ϕ > 0} at two points, and similarly {(0, ξ ), ξ > 0}, by Remark 2.1(i) on page 211. For suitable values of α, δ, N , we can show that cycles cannot exist, by using Bendixson’s criterion or the Poincaré map. Writing (S) under the form (2–49)

y 0 = f 1 (y, Y ),

Y 0 = f 2 (y, Y ),

we obtain (2–50)

∂ f1 ∂ f2 (y, Y ) + (y, Y ) = 2δ − N − ε|Y |(2− p)/( p−1) . ∂y ∂Y

For example, as a direct consequence of Bendixson’s criterion, if ε(δ − N /2) < 0, we find again the nonexistence of any cycle in R2 , which was obtained in Proposition 2.11. Now we consider cycles in Q1 . First we extend to system (S) a general property of quadratic systems, proved in [Chicone and Tian 1982], stating that there cannot exist a closed orbit surrounding a node point. Note that the restriction of our system to Q1 is quadratic if p = 32 . Theorem 2.18. Let δ < N and ε(δ − α) < 0. When M` is a node point, there is no cycle or homoclinic orbit in Q1 . Proof. We use the linearization (2–27), (2–28), (2–29). Consider the line L with equation Ay + Y = 0, where A is a real parameter. The points of L are in Q1 whenever −(δ`) p−1 < Y and −` < y. As in [Chicone and Tian 1982], we study the orientation of the vector field along L: we find Ay 0 + Y 0 = (εν(α)A2 + (N + ν(α))A + εα)y − (A + ε)9(Y ). Using (2–31), apart from the case ε = 1, α = N = α1 , we can find an A such that εν(α)A2 + (N + ν(α))A + εα = 0, and A + ε 6= 0. Moreover 9(Y ) ≥ 0 on L ∩ Q1 ; indeed, ( p − 1)9 0 (t) = ((δ`) p−1 + t)(2− p)/( p−1) − t (2− p)/( p−1) , so 9 has a minimum on (−(δ`) p−1 , ∞) at 0, and hence is nonnegative on this interval. Then the orientation of the vector field does not change along L ∩ Q1 ; in particular no cycle can exist in Q1 ; and similarly no homoclinic trajectory can exist. In the case ε = 1, α = N = α1 , Y ≡ y ∈ [0, `) defines the trajectory Tr , corresponding to the solutions given by (1–6) with K > 0, and again no cycle can exist in Q1 : it would intersect Tr .

p -LAPLACE EQUATION


229

Next we prove the nonexistence of cycles on one side of the Hopf bifurcation point: Theorem 2.19. Assume δ < N and ε(δ − α) < 0 < ε(δ − N /2). If ε(α − α ∗ ) ≥ 0, there exists no cycle or homoclinic orbit in Q1 . Proof. M` is a source or weak source if ε = 1, and a sink or weak sink if ε = −1. Suppose there exists a cycle in Q1 . Then any trajectory starting from M` at −ε∞ has a limit cycle in Q1 , which is attracting at ε∞. Such a cycle is not unstable (if ε = 1) or not stable (if ε = −1); in other words the Floquet integral on the period [0, P] is nonpositive if ε = 1 and nonnegative if ε = −1. From (2–50) we then get Z P ∂ f1 ∂ f2 (2–51) ε (y, Y ) + (y, Y ) dτ ∂y ∂Y 0 Z P 1 (2− p)/( p−1) = |2δ − N | − Y dτ ≤ 0. p−1 0 Now, from (2–28), Z 0=δ

P

ydτ − ν(α)

0=α

P

P

Z Y dτ −

0

0

Z

P

Z

ydτ + (δ − N − ν(α))

0

9(Y ) dτ,

0 P

Z

P

Z Y dτ −

0

9(Y ) dτ.

0

Moreover, since 9 is nonnegative, Z P Z P Z p 0 (N − δ) P ydτ = − Y dτ > 0; 9(Y ) dτ = − p 0 α−δ 0 0 0 and since y 0 = δy − Y 1/( p−1) , Z P Z 1/( p−1) Y dt = δ 0

P

y dt < δ`P.

0

From this, (2–51), and Jensen’s inequality, it follows that Z P ( p − 1)|2δ − N | ≤ Y (2− p)/( p−1) dτ 0 Z P 2− p εδ(N − δ) Y 1/( p−1) dτ < (δ`)2− p = . ≤ P p−1 α−δ 0 Hence ε(α − α ∗ ) < 0, a contradiction. Next, suppose that there is an homoclinic orbit. From [Hubbard and West 1995, Theorem 9.3, p. 303] we see that the saddle connection is repelling if ε = 1 and attracting if ε = −1, because the sum of the eigenvalues µ1 , µ2 of the linearized problem at (0, 0) is 2δ − N . That means that the solutions just inside it spiral toward the loop near −ε∞. Because M` is a

230


source or weak source or sink or weak sink, such solutions have a limit cycle that is attracting at ε∞. As before, we reach a contradiction. Finally we get the nonexistence of cycles in nonobvious cases, where we have shown that any solution has at most one or two zeros. Theorem 2.20. Assume δ < N and ε(δ − α) < 0 < ε(δ − N /2). If ε = 1 and α ≤ η, or ε = −1 and − p 0 ≤ α < 0, there exists no cycle and no homoclinic orbit in Q1 . Proof. (i) Suppose there exists a cycle. There are two possibilities: Suppose ε = 1 and α ≤ η. M` is a sink since α < α ∗ , so any trajectory converging to M` at ∞ has a limit cycle O in Q1 , attracting at −∞. Let (y, Y ) describe the orbit O, of period P. Then O is not stable, so the Floquet integral is nonnegative, and from (2–51), Z P 1 Y (2− p)/( p−1) dτ ≥ 0. 2δ − N − p−1 0 Otherwise y is bounded from above and below; thus the function yα , defined by (2–3) with d = α, satisfies limτ →−∞ yα = 0 and limτ →∞ yα = ∞; moreover yα has only minimal points, from (2–35), since α ≤ η; thus yα0 > 0 on R. From (2–5) and (2–4) with d = α, 1 α(η − α)yα yα00 α(η − α)yα + η − 2α + = Y (2− p)/( p−1) = > η − α. 1/( p−1) 0 0 yα p−1 yα αyα − Yα Upon integration over [0, P], this implies η − 2α + 2δ − N > η − α, which is impossible, since δ − N + δ − α < 0. Alternatively, suppose ε = −1 and − p 0 ≤ α < 0. M` is a source since α ∗ < α, and any trajectory converging to it at −∞ has a limit cycle O0 attracting at ∞. Let (y, Y ) describe the orbit O0 , of period P. Then O0 is not unstable, so the Floquet integral is nonpositive, hence Z P 1 (2δ − N + Y (2− p)/( p−1) ) dτ ≤ 0. p−1 0 Moreover Y is bounded from above and below; thus Yα , defined by (2–3) with d = α, satisfies limτ →−∞ Yα = ∞, limτ →∞ Yα = 0. And Yα has only minimal points, by (2–36), since − p 0 ≤ α < 0; thus Yα0 < 0 on R. From (2–6) and (2–4) we get Yα00 1 ( p − 1)2 (η − α)( p 0 + α)Yα 0 (2− p)/( p−1) + ( p − 1)(η − 2α − p ) − Y = Yα0 p−1 Yα0 < −( p − 1)( p 0 + α). Upon integration over [0, P], this implies ( p−1)(η−2α− p 0 )+2δ−N < −( p−1) × ( p 0 + α), which means pδ + ( p − 1)|α| < 0; but this is false.


p -LAPLACE EQUATION

231

(ii) Suppose there exists an homoclinic orbit. Since δ < N , the origin is a saddle point, so Tr is the only trajectory starting from (0, 0) in Q1 , and there exists a unique trajectory Ts converging to (0, 0), lying in Q1 for large τ , having infinite slope at (0, 0), and satisfying lim r →0 r η w = c > 0. If ε = 1, then Tr satisfies limτ →−∞ e−ατ yα = a > 0, so limτ →−∞ yα = 0; also yα has only minimal points, so it is increasing and positive; and Ts satisfies limτ →∞ e(η−α)τ yα = c > 0. If α < η, then limτ →∞ yα = 0, thus Tr 6= Ts . If α = η, Ts is given explicitly by (1–7), that means yα is constant, thus again Tr 6 = Ts . If ε = −1, then Ts satisfies limτ →−∞ e(η−α)( p−1)τ Yα > 0, because limτ →−∞ ζ = η; so limτ →∞ Yα = 0. Moreover Yα has only minimal points, and hence is increasing and positive; otherwise Tr satisfies limτ →−∞ e−(α( p−1)+ p)τ Yα = −aα/N > 0, by (2–33). If α > − p 0 , we get limτ →∞ Yα = 0, which implies Tr 6= Ts . If α = − p 0 , then Tr is given explicitly by (1–8); in other words Yα is constant, and again Tr 6 = Ts . Boundedness of cycles. When there do exist cycles, except for a few cases, we cannot prove their uniqueness, but we can show: Theorem 2.21. When nonempty, the set C of cycles of system (S) is bounded in R2 . Proof. Suppose there exists a cycle O in R2 . By Propositions 2.5, 2.7, 2.10, 2.11 and Theorem 2.20, this can happen only in four cases: ε = 1, N < α < δ; ε = 1, N < δ = α; ε = 1, max(δ, N , η) < α, N /2 < δ; ε = −1, δ < N /2, α < − p 0 . In the first case, C is bounded and lies in (−`, `)×(−(δ`) p−1 , (δ`) p−1 ), by Proposition 2.10. In the other cases we use the energy function W . Let (y, Y ) describe the trajectory O. Then W is periodic, and its maximum and minimum points are precisely the points of the curve L. Indeed if W 0 (τ1 ) = 0 and the point (y(τ1 ), Y (τ1 ) is not on L, it lies on the curve M defined in (2–11); hence y 0 (τ1 ) = 0 and y 00 (τ1 ) 6= 0, since O is not just a stationary point. Therefore (δy − |Y |(2− p)/( p−1) Y )(|δy|) p−2 δy − Y ) > 0 near τ1 ; then W 0 has constant sign, and τ1 is not a maximum or a minimum. In this way we obtain estimates for W independently of the trajectory: max |W (τ )| = M = max |W(y, Y )|. τ ∈R

(y,Y )∈L

At the maximal points τ of y, one has |Y (τ )|(2− p)/( p−1) Y (τ ) = δy(τ ), so W (τ ) =

ε(δ − N )δ p−1 α−δ 2 |y(τ )| p + y (τ ). p 2

By the Hölder inequality, y is bounded by a constant independent of the trajectory, and 0 |Y | p |2δ − N |δ p−1 p |α − δ| 2 |y| + y + M. ≤ δyY + p0 p 2 Thus Y is also uniformly bounded, and C is bounded.

232


3. The case ε = 1, α < δ or α = δ < N Lemma 3.1. Assume ε = 1 and −∞ < max(α, N ) < δ(α 6= 0). In the phase plane (y, Y ), there exist (i) a trajectory T1 converging to M` at ∞, such that y is increasing as long as it is positive; (ii) a trajectory T2 in Q1 ∪ Q4 converging to M` at −∞, and unbounded at ∞, with limτ →∞ ζ = α; (iii) a trajectory T3 converging to M` at −∞, such that y has at least one zero; (iv) a trajectory T4 in Q1 , converging to M` at ∞, with limτ →ln Rw Y/y = 1; (v) trajectories T5 in Q1 ∪ Q4 unbounded at ±∞, with lim ζ = α

τ →∞

and lim Y /y = 1. τ →ln Rw

Proof. Here the system (S) has three stationary points, defined by (2–26). The point (0, 0) is a source, and the point M` is a saddle point. The eigenvalues satisfy λ1 < 0 < λ2 < δ. The eigenvectors u 1 = (−ν(α), λ1 − δ) and u 2 = (ν(α), δ − λ2 ) form a positively oriented basis, and u 1 points toward Q3 , while u 2 points toward Q1 . There exist four particular trajectories converging to M` at ±∞, namely: •

T1 converging to M` at ∞, with tangent vector u 1 ; then y < ` and Y < (δ`) p−1

and y 0 > 0 near ∞; as above, y cannot have a local minimum, so y 0 > 0 whenever y > 0. •

T2 converging to M` at −∞, with tangent vector u 2 ; then y 0 > 0 near −∞.

If y has a local maximum at some τ , then y 00 (τ ) ≤ 0, so that y(τ ) ≤ ` from (2–16), which is impossible. Then y is increasing on R and limτ →∞ y = ∞, and limτ →∞ ζ = α from Proposition 2.8. In particular T2 stays in Q1 if α > 0, and enters Q4 if α < 0. •

T3 converging to M` at −∞, with tangent vector −u 2 ; then y 0 < 0 near −∞.

If y has a local minimum at some τ , then y(τ ) ≥ `, which is still impossible. Thus y is decreasing at long as the trajectory stays in Q1 . It cannot stay in it, because it cannot converge to (0, 0). It cannot enter Q4 by Remark 2.1(i) on page 211. Then it enters Q2 and y has at least one zero. •

T4 converging to M` at ∞, with tangent vector −u 1 ; then y 0 < 0 near ∞. As

above, y cannot have a local maximum, it is decreasing and limτ →ln Rw y = ∞. From Proposition 2.8, y cannot be defined near −∞, hence Rw > 0 and limτ →ln Rw Y /y = 1. For any trajectory T in the domain delimited by T2 and T4 , the function y is positive, and T cannot converge to M` at ∞, and y is monotone for large τ from


p -LAPLACE EQUATION

233

Proposition 2.7, because α < δ; thus limτ →∞ ζ = α from Proposition 2.8, and y is not defined near −∞, and T is of type (5). We now study the various global behaviors, according to the values of α. The results are expressed in terms of w. α≤ N 0 if α < N , lim r →∞ r δ |w| = ` if α = N . And w(r ) = `r −δ is also a solution. There exist solutions satisfying any one of these characterizations: (1) (only if α < N ) w is positive, lim r →0 r η w = c > 0, if N ≥ 2 (and (2–41) holds with a > 0 > b if N = 1), and lim r →∞ r δ w = `; (2) w is positive, lim r →0 r δ w = `, lim r →∞ r α w = L > 0; (3) w has precisely one zero, lim r →0 r δ w = `, lim r →∞ r α w(r ) = L < 0; (4) w is positive, Rw > 0, lim r →∞ r δ w = `; (5) w is positive, Rw > 0, lim r →∞ r α w = L > 0; (6) w has one zero, Rw > 0, and lim r →∞ r α w = L 6= 0; (7) (only if α < N ) w is positive, lim r →0 r η w = c > 0 if N ≥ 2 (and (2–41) holds with a > 0 > b if N = 1), and lim r →∞ r α w = L > 0; (8) w has one zero, with lim r →0 r η w = c > 0 if N ≥ 2 (and (2–41) holds with a > 0 > b if N = 1), and limr →∞ r α w = −L < 0; (9) N = 1, w > 0 and (2–41) holds with a ≥ 0, b > 0 and lim r →∞ r α w = L. Up to symmetry, all the solutions of (E w ) are as above.

α=1< N

α=2= N

Figure 1. Theorem 3.2: N = 2 < δ = 3.

234


Proof. (i) We first assume that α 6= N , and refer to Figure 1, left. The trajectory Tr starts in Q1 for α > 0, in Q4 for α < 0, and y stays positive. Then limτ →∞ y = ∞, and limτ →∞ ζ = α, and limr →∞ r α w = L > 0, by Propositions 2.10 and 2.13, since α < N . Moreover y is increasing: indeed if it has a local maximum, at this point y ≤ `, and then y has no local minimum, since at such a point y ≥ `, so that y cannot tend to ∞. Then Tr stays in Q1 , and Y is increasing from 0 to ∞. Indeed each extremal point τ of Y is a local minimum, from (2–17). If α < 0, in the same way, then Y is decreasing from 0 to −∞, and Tr stays in Q4 . First we follow the trajectory T1 : it does not intersect Tr , and cannot enter Q2 by Remark 2.1(i). Thus y stays positive and increasing. It cannot enter Q4 , seeing that it does not meet Tr if α > 0, or (by the same remark) if α < 0. Thus T1 stays in Q1 , and (y, Y ) converges necessarily to (0, 0). If N ≥ 2, then limτ →−∞ ζ = η, lim r →0 r η w = c > 0 from Proposition 2.8 and 2.9. If N = 1, since T1 stays in Q1 , then necessarily limτ →−∞ ζ = 0, thus (2–41) holds with a > 0 > b. Next we follow T3 : here y has a zero, which is unique by Proposition 2.5, since α < N . Then y < 0, and limτ →∞ y = −∞, lim r →∞ r α w = −L < 0 by Propositions 2.8 and 2.9. T3 stays in Q2 if α < 0, or goes from Q2 into Q3 if α > 0. Trajectories T2 , T4 , T5 of Lemma 3.1 yield solutions w of type (2), (4), (5). For any trajectories T6 in the domain delimited by T3 , T4 , y has one zero, and lim r →∞ r α w = L 6= 0; and w is of type (6). The solutions of type (7) correspond to the trajectories T in the domain delimited by Tr , T1 , T2 . Indeed limτ →∞ y = ∞, and lim r →∞ r α w = L > 0. And limτ →−∞ y = 0. If N ≥ 2, then limτ →−∞ ζ = η, lim r →0 r η w = c > 0, from Proposition 2.8 and 2.9. If N = 1, T cannot meet Tr , thus necessarily limτ →−∞ ζ = 0, and (2–41) holds with a > 0 > b. Up to a change of w into −w, the solutions of type (8) and (9) correspond to the trajectories in the domain delimited by −Tr , T1 , T3 . Indeed they satisfy limτ →∞ y = −∞, and lim r →∞ r α w = L < 0; and limτ →−∞ y = 0. If N ≥ 2, then lim r →0 r η w = c > 0 and w has a zero. If N = 1, either (2–41) holds with a = 0 > b and w stays negative, or a < 0, b < 0 and w has a zero. Such solutions exist from Theorem 2.2. By symmetry, all the solutions are described. (ii) Now assume α = N (Figure 1, right). Then M` belongs to the line y = Y , and u 1 = (−δ/( p − 1), −δ/( p − 1)) has the same direction. Moreover JN is constant, which means y − Y = Ce(δ−N )τ , with C ∈ R. The solutions corresponding to C = 0 satisfy y ≡ Y , thus T1 = Tr = {(ξ, ξ ) : ξ ∈ [0, `)}, corresponding to the regular Barenblatt solutions. And T4 = {(ξ, ξ ) : ξ > `} yields the solutions defined by (1–6) for K < 0. All other solutions exist as before, apart from type (7).


p -LAPLACE EQUATION

235

Note. The trajectory T1 is the only one joining the stationary points (0, 0) and M` . Hence, for α < N , solutions w of type (1) are unique, up to the scaling mentioned in the note on page 210. Solutions of types (2), (4), and (5) are also unique. N k` , then Ck has two unbounded connected components. If 0 < k < k` , Ck has three connected components, of which one is bounded. If k = k` , Ck` is connected with a double point at M` . If k = 0, one of the three connected components of C0 is {(0, 0)}. If k < 0, Ck has two unbounded connected components. Proof. The energy k` of the statement is positive. Also (y, Y ) ∈ Ck if and only if F(y) = k − G(y), where F, G are defined in (2–45). By symmetry we can reduce 0 the study of Ck to the set y > 0. Let ϕ(s) = |s| p / p 0 − s + 1/ p for any s ∈ R, and set θ = Y /(δy) p−1 . Then (2–44) reduces to ϕ(θ) = (k − G(y))/(δy) p . The function ϕ is decreasing on (−∞, 1) from ∞ to 0, and increasing on (1, ∞) from 0 to ∞. Let ψ1 be the inverse of the restriction of ϕ to (−∞, 1] and ψ2 the inverse of the restriction of ϕ to [1, ∞), both defined on [0, ∞). For any y > 0, y ∈ Ck ⇐⇒ Y = 81 (y) < (δy) p−1 or Y = 82 (y) ≥ (δy) p−1 , where (3–1)

8i (y) = (δy) p−1 ψi

k − G(y) (δy) p

for i = 1, 2,

81 lies below M whereas 82 lies above M, and 81 , 82 ∈ C 1 ((0, ∞)). The function G has a maximal point at y = `, and G(`) = k` . Using symmetry we see that either k > k` and y ranges over R, in which case Ck has two unbounded connected components; or 0 < k < k` and Ck has three connected components, one of which, Cbk , is bounded; or k = k` and Ck` is connected with a double point at M` ; or yet k = 0 and one of the three connected components of C0 is {(0, 0)}; or k < 0 and Ck

236


has two unbounded connected components. The unbounded components satisfy 0 0 lim|y|→∞ Y/y 2/ p = ±( p 0 (δα)/2)1/ p , by (3–1). The zeros of 8i0 are contained in N = (y, Y ) ∈ R2 : y > 0, δY = −(δ − α)y + (2δ − N )(δy) p−1 , and N lies above M as long as y < `. We now describe Cbk when 0 < k ≤ k` . The function 81 is increasing on a 0 segment [0, y] such that y < `, and 81 (0) = −(kp 0 )1/ p and (y, 81 (y)) ∈ M, with an infinite slope at this point; 82 is increasing on some interval [0, y˜ ) such that 0 ( y˜ , 82 ( y˜ )) ∈ N and then decreasing on ( y˜ , y], and 82 (0) = (kp 0 )1/ p and 82 (y) = 81 (y). By symmetry with respect to (0, 0), the curve Cbk is completely described. Next consider Ck` for y > 0: the function 82 is increasing on [0, ∞) from 0 ( p 0 k` )1/ p to ∞, and 82 (`) = (δ`) p−1 ; the function 81 is increasing on some interval [0, yˆ ) such that ( yˆ , 81 ( yˆ )) ∈ N, so yˆ > `; and ( yˆ , 81 ( yˆ )) is below M, and 81 (`) = (δ`) p−1 , and 81 is decreasing on ( yˆ , ∞), with lim y→∞ 81 = −∞. Setting Ck` ,1 = {(y, 81 (y)) : y > `} and Ck` ,2 = {(y, 82 (y)) : y > `}, one has Ck` = Cbk` ∪ ±Ck` ,1 ∪ Ck` ,2 . Theorem 3.4. Assume ε = 1 and N < α < δ. Then w(r ) = `r −δ is a solution of (E w ). (i) If α ≤ α ∗ , any solution of (E w ) has at most a finite number of zeros. (ii) There exist αˇ such that max(N , α ∗ ) < αˇ < δ, such that if α > α, ˇ in the phase plane (y, Y ), there exists a cycle surrounding (0, 0). (iii) Let α be such that there exists no such cycle. Then all regular solutions have a finite positive number of zeros and limr →∞ r α w = L r 6= 0 or lim r →∞ r δ w = ±`. There exist solutions of types (2)–(6) of Theorem 3.2, and solutions such that 0 (1 ) (only if L r 6= 0) lim r →0 r δ w = `, and lim r →0 r η w = c 6= 0 (or (2–41) holds if N = 1); 0 (7 ) lim r →0 r η w = c 6= 0 (or (2–41) holds if N = 1) and lim r →∞ r α w = L 6= 0. (iv) For any α such that there exists such a cycle, there exist solutions w which oscillate near 0 and ∞, and r δ w is periodic in ln r . All regular solutions w oscillate near ∞, and r δ w is asymptotically periodic in ln r . There exist solutions of types (2), (4), (5), and solutions 00 (1 ) with precisely one zero, Rw > 0, and limr →∞ r δ w = `; (300 ) such that lim r →0 r δ w = `, and oscillating near ∞; (9) such that lim r →0 r η w = c 6= 0 (or (2–41) holds if N = 1) and oscillating near ∞; (10) with precisely one zero, Rw > 0, and lim r →∞ r α w = L 6= 0; (11) with Rw > 0 and oscillating near ∞.


p -LAPLACE EQUATION

237

Proof. There always exist solutions of type (2), (4), and (5), by Lemma 3.1. (i) Assume α ≤ α ∗ (see Figure 2, left). Consider any trajectory T. Suppose y has infinitely many zeros near ±∞. From Proposition 2.10, T is contained in the set D = {(y, Y ) ∈ R2 : |y| < `, |Y | < (δ`) p−1 }

near ±∞. Then T is bounded near ±∞, hence the limit set at ±∞ is contained in D. But M` 6∈ D, and (0, 0) is a source and a node point, so it cannot be in the limit set 0 at ∞. From the Poincaré–Bendixson theorem, 0 is a closed orbit, so that there exists a cycle. Moreover, from (2–25), (2–49) and (2–50), ∂ f1 ∂ f2 1 (y, Y ) + (y, Y ) = (D (2− p)/( p−1) − |Y |(2− p)/( p−1) ); ∂y ∂Y p−1 thus, by Bendixson’s criterion, the set {|Y | < D} contains no cycle. Now note that α ≤ α ∗ ⇐⇒ (δ`) p−1 ≤ D.

(3–2)

Then there is no cycle in D, and we reach a contradiction. (ii) Now assume α > max(N , α ∗ ). The curve L intersects M at (δ −1 D 1/( p−1) , D). Then SL ∩ M = (δ −1 (θ D)1/( p−1) , θ D) : θ ∈ [0, 1] ; and D < (δ`) p−1 by (3–2), so SL does not contain M` . We can find k1 > 0 small enough that Cbk1 is interior to SL . Next we search for k ∈ (0, k` ) such that L is in the domain delimited by Ckb . By symmetry we only consider the points of L such that 0 y ≥ 0. In any case for any point of L we have |δy| p + |Y | p ≤ M = (2(2δ − N ))δ , by (2–23) and by convexity. By a straightforward computation this implies that

α = 2.41

α = 2.42

Figure 2. Theorem 3.4: ε = 1, N = 2 < α < δ = 3.

238


W(y, Y ) ≤ K M, where K = max(2/ p 0 , (3δ − N )/δp). Let αˇ = α(δ, ˇ N ) be given

by K M = k` . This means that δ − αˇ =

δ − N 1/δ δ p−1 (δ − N ) . 2K δ 2− p 2(2δ − N )

If α > α, ˇ there exists k2 < k` such that L is contained in the set {(y, Y ) ∈ R2 : W(y, Y ) < k2 }, which has three connected components; because SL is connected, it is contained in the interior to Ck b . Then the domain delimited by Cbk1 and Cbk2 is bounded and 2 forward invariant. It does not contain any stationary point, and so it contains a cycle, by the Poincaré–Bendixson theorem (see Figure 2, right). (iii) Let α be such that there exists no cycle. Since N < α, all regular solutions y have at least one zero. They have a finite number of zeros. For if not, (y, Y ) is bounded near ∞, so it has a limit cycle. Then either limτ →∞ y = ±∞ and limτ →∞ ζ = α > 0, so that the trajectory Tr ends up in Q1 ∪Q3 and lim r α w = L r 6= 0, or else limτ →∞ y = ±` and lim r →∞ r δ w = ±`. T3 cannot meet Tr or −Tr , thus y has a unique zero, and limτ →∞ y = −∞, and limτ →∞ ζ = α. The same happens for the trajectories T6 in the domain delimited by T3 , T4 . Thus there exist solutions of types (3) and (6). Suppose L r 6= 0 and consider T1 : the trajectories Tr , −Tr , T1 have a last intersection point at time τ0 with the half-axis {y = 0, Y < 0} at some points Pr , Pr0 , P1 , and P1 ∈ [Pr , Pr0 ]. The domain delimited by Tr , −Tr and [Pr , Pr0 ] is bounded and backward invariant, by Remark 2.1(i) on page 211. Then T1 stays in it for τ < τ0 , it has a finite number of zeros, and converges to (0, 0) near −∞; thus w is of type (10 ). If N ≥ 2, then limτ →∞ ζ = η, so that y has at least one zero. Since (0, 0) is a source, there exist other solutions converging to (0, 0) near −∞, they have a finite number of zeros, and limτ →∞ ζ = α, and w is of type (70 ). (iv) Let α such that there exists a cycle, thus Tr has a limit cycle O. Consider again T1 . Since M` 6∈ SL , the function W is decreasing near ∞, so that W (τ ) > k` ; thus T1 is exterior to Cbk` for large τ , in the domain exterior to Cbk` delimited by Ck` ,1 and −Ck` ,2 ; and it cannot cut Ck` . Moreover y is decreasing at long as y > 0, then T1 enters Q4 as τ decreases. It cannot stay in it, because it would converge to (0, 0), which is impossible. Then y has at least one zero, and T1 enters Q3 . It stays in it, since it cannot cross −Ck` ,2 . Thus y has a unique zero, and limτ →−∞ y = −∞, and Rw > 0 from Proposition 2.8, because T1 cannot converge to (0, 0) at −∞, and w is of type (100 ). Next consider T3 . Here W is decreasing near −∞, hence W (τ ) < k` ; thus T3 is in the interior of Cbk` near −∞. Now the domain delimited by Cbk1 and Cbk` is


p -LAPLACE EQUATION

239

forward invariant, thus T3 stays in it; then it is bounded, and has a limit cycle at ∞, and w is of type (300 ). The solutions of type (9) correspond to trajectories T in the domain delimited by O, and distinct from Tr . Indeed T is bounded, in particular the limit-set at −∞ is (0, 0), or a closed orbit. But T cannot intersect Tr . Then T converges to (0, 0) near −∞. The solutions of type (10) correspond to a trajectory T in the domain delimited by T1 ∪ T2 (or its opposite): indeed y has constant sign near ∞ and near ln Rw , and lim r →∞ r α w = L 6= 0, and Rw > 0, from Proposition 2.8. Then T starts in Q3 , and ends up in Q1 ; and y has at most one zero, because at such a point y 0 = −|Y |1/( p−1) Y > 0, thus it has precisely one zero. Solutions of type (11) correspond to a trajectory T in the domain delimited by T1 , T4 , −T1 , −T4 . Then y cannot have constant sign near ∞: indeed this implies lim ζ = α > 0; this is impossible since the line Y = y is an asymptotic direction for T1 , T4 . Thus T is bounded near ∞, and it has a limit cycle at ∞. Near −∞, y a constant sign, because T cannot meet T3 ; and Rw > 0 from Proposition 2.8, and T has the same asymptotic direction Y = y as T1 , T4 . Note. From numerical studies, we conjecture that αˇ is unique, and the number of zeros of w increases with α in the range (N , α); ˇ and moreover there exists α1 = N < α2 < · · · < αn < αn+1 < . . . , such that regular solutions have n zeros for any α ∈ (αn , αn+1 ), with lim r →∞ r α w = L r 6= 0, and n + 1 zeros for α = αn+1 , with lim r →∞ r δ w = ±`. α ≤ δ ≤ N, α 6= N Here (0, 0) is the only stationary point, and N ≥ 2. Theorem 3.5. Assume ε = 1 and −∞ < α ≤ δ ≤ N , α 6= 0, N . Then all regular solutions of (E w ) have constant sign, and the positive ones satisfy lim r →∞ r α w(r ) = L > 0 if α 6= δ, or (2–39) holds if α = δ. All the other solutions have a reduced domain (Rw > 0). Among them, there exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →∞ r η w = c 6= 0 if δ < N , or lim r →∞ r N (ln r )(N +1)/2 w = % defined in (2–40) if δ = N ; (2) w is positive, lim r →∞ r α w = L > 0 if α 6= δ, or (2–39) holds if α = δ; (3) w has one zero, and lim r →∞ r α w = L 6= 0 if α 6= δ, or (2–39) holds if α = δ. Up to symmetry, all the solutions are as above. Proof. Any solution has at most one zero, by Proposition 2.5. The trajectory Tr starts in Q4 if α < 0 (Figure 3, left) and in Q1 if α > 0 (Figure 3, right). Moreover y

240


α = −2

α=2

Figure 3. Theorem 3.5: ε = 1, α < δ = 3 < N = 4. stays positive, and limτ →∞ y = ∞ and limτ →∞ ζ = α, by Proposition 2.13. Then lim r →∞ r α w(r ) = L > 0 if α < δ, or (2–39) holds if α = δ, from Proposition 2.9. Moreover y is increasing: indeed it has no local maximum from (2–16). Thus Tr does not meet M, and so stays below M. If α > 0, then Tr stays in Q1 , and Y is increasing from 0 to ∞; indeed each extremal point τ of Y is a local minimum, by (2–17). Likewise, if α < 0 the function Y is decreasing from 0 to −∞, and Tr stays in Q4 . The only solutions y defined on (0, ∞) are the regular ones, by Proposition 2.8. For any point P = (ϕ, (δϕ) p−1 ) ∈ R2 with ϕ > 0, in other words on the curve M, the trajectory T[P] intersects M transversally: the vector field is (0, −(N −α)ϕ). Moreover the solution going through this point at time τ0 satisfies y 00 (τ0 ) > 0 from (E y ), then τ0 is a point of local minimum. From (2–16), τ0 is unique, so that it is a minimum. Then y > 0, limτ →∞ ζ = α, limτ →ln Rw Y /y = 1, and T[P] stays in Q1 if α > 0, or goes from Q1 into Q3 if α < 0. The corresponding w is of type (2). For any point P = (0, ξ ), ξ > 0, the trajectory T[P] goes through P from Q1 into Q2 , by Remark 2.1(i). Then y has only one zero, and as above, it is decreasing on R and limτ →∞ y = −∞, and limτ →∞ ζ = α, limτ →ln Rw Y/y = 1. Thus T[P] starts in Q1 , then stays in Q2 if α < 0, and enters Q3 and stays in it if α > 0. The corresponding w is of type (3). It remains to prove the existence of a solution of type (1). If δ < N , then (0, 0) is a saddle point. There exists a trajectory T1 converging to (0, 0) at ∞, with y > 0, and limτ →∞ ζ = η > 0, thus in Q1 near ∞, with y 0 < 0. As above, y has no local maximum, it is increasing, so that y > 0. If δ = N , we consider the sets A = {P ∈ (0, ∞) × R : T[P] ∩ M 6 = ∅}, B = {P ∈ (0, ∞) × R : T[P] ∩ {(0, ξ ) : ξ > 0} 6 = ∅}.


p -LAPLACE EQUATION

241

They are nonempty, and open, because the intersections are transverse. Since Tr is below M, the sets A and B are contained in the domain R of Q1 ∪ Q2 above Tr , and A ∪ B 6 = R. As a result there exists at least a trajectory T1 above Tr , which does not intersects M and the set {(0, ξ ) : ξ > 0}. The corresponding y is monotone. Suppose that y is increasing, then limτ →−∞ y = 0; it is impossible since T1 6 = Tr . Then y is decreasing, and limτ →∞ y = 0. In any case w is of type (1), by Propositions 2.8 and 2.9. All the solutions are described, because any solution has at most one zero, and at most one extremum point. And T1 is unique when δ < N. 4. The case ε = −1, δ < α N 0 for any a > 0 and lim r →0 w0 = b < 0, b 6= b(a) if N = 1), and Sw < ∞; (8) w has one zero and the same behavior; (9) (only if N = 1) w is positive, lim r →0 w = a > 0, and lim r →0 w0 = b > 0, and Sw < ∞. Up to symmetry, all solutions are as above. Proof. Here we still have three stationary points, (0, 0) is a source and M` a saddle point (see Figure 4). By Propositions 2.5 and 2.14, all regular solutions have constant sign and satisfy Sw < ∞. Also, Tr stays in Q4 by Remark 2.3, and limτ →ln Sw Y/y = −∞ by Proposition 2.15. Since α > 0, any solution y has at most one zero, by Proposition 2.5, and y is monotone near ln Sw (finite or not) and near −∞, by Proposition 2.7. In the linearization near M` the eigenvectors u 1 = (ν(α), λ1 −δ) and u 2 = (−ν(α), δ−λ2 ) form a positively oriented basis, where

242


N = 2, α = 4

N = 1, α = 5

Figure 4. Theorem 4.1: ε = −1, N < δ = 3 < α. now ν(α) < 0 and λ1 < δ < λ2 ; thus u 1 points toward Q3 and u 2 points toward Q4 . There exist four particular trajectories converging to M` near ±∞, namely: •

T1 converging to M` at ∞, with tangent vector u 1 . Here y is increasing near

∞, and as long as y > 0; indeed, if there exists a minimal point τ , (E y ) shows that y(τ ) > `. And T1 stays in Q1 on R, by Remark 2.1(i) on page 211. Therefore T1 converges to (0, 0) at −∞, and w is of type (1), where b(a) is a function of a, by the note on page 210. •

T2 converging to M` at −∞, with tangent vector u 2 . Here again y 0 > 0 as

long as y > 0. Also Y 0 < 0 near −∞, and Y is decreasing as long as Y > 0: if there exists a minimal point of Y in Q1 , (E Y ) shows that Y (τ ) > (δ`) p−1 . But (y, Y ) cannot stay in Q1 , as this would imply limτ →∞ y = ∞, which is impossible by Proposition 2.8. Thus T2 enters Q4 at some point (ξ2 , 0) with ξ2 > 0 and stays in it since y 0 > 0. Thus Sw < ∞ and limτ →∞ Y /y = −1, and w is of type (2). •

T3 converging to M` at −∞, with tangent vector −u 2 . Here again y 0 < 0 as long as y > 0. And Y 0 > 0 as long as Y > 0; thus Y 0 > 0 on R. Then again (y, Y ) cannot stay in Q1 , so y has a unique zero, and T3 enters Q2 at some point

(0, ξ3 ) with ξ3 > 0 and stays in it. Hence Sw < ∞ and limτ →∞ Y /y = −1, and w is of type (3). •

T4 converging to M` at ∞, with tangent vector −u 1 . In the same way, y is decreasing near ∞, and y is everywhere decreasing: if there exists a maximal point τ , then y(τ ) < ` by (E y ). Then Y stays positive, thus T4 stays in Q1 . By Proposition 2.8, limτ →−∞ y = ∞ and limτ →−∞ ζ = α, so w is of type (4).

Next we describe all the other trajectories T[P] with one point P in the domain

R above Tr ∪ (−Tr ).


p -LAPLACE EQUATION

243

If P = (ϕ, 0) with ϕ > ξ2 , then T[P] stays in Q4 after P, because it cannot meet T2 ; before P it stays in Q1 , by Remark 2.1(i). Thus again Sw = ∞, and limτ →−∞ ζ = α > 0, and y has a unique minimal point, and w is of type (5). For any P is in the domain delimited by T2 , T4 , the trajectory T[P] is of the same type. If P = (0, ξ ) with ξ > ξ3 , then T[P] stays in Q2 after P, in Q1 before P, since it cannot meet T2 , T4 . Then limτ →−∞ ζ = α > 0, and Sw = ∞, and w is of type (6). If P is in the domain delimited by T3 , T4 , then T[P] is of the same type. If P = (ϕ, 0) with ϕ ∈ (0, ξ2 ), then T[P] stays in Q4 after P, in Q1 before P; it cannot meet Tr , thus Sw < ∞; and T[P] converges to (0, 0) in Q1 at −∞; thus w is of type (7), by Theorem 2.2. If P is in the domain delimited by T1 , T2 , Tr , then T[P] is of the same type. If P = (0, ξ ) for some ξ ∈ (0, ξ3 ), then T[P] stays in Q2 after P, in Q1 before P; and T cannot meet −Tr , so that Sw < ∞. Then T[P] converges to (0, 0) in Q1 at −∞, and w is of type (8). If P lies in the domain delimited by T1 , T3 and −Tr , either y has one zero, and T[P] is of the same type; or y < 0 on R, and y 0 = δy − Y 1/( p−1) < 0. Hence Sw < ∞ and T[P] converges to (0, 0) in Q2 at −∞. It implies N = 1 (see Figure 4, right), and −w is of type (9), by Propositions 2.8 and 2.9; and such a solution does exist, by Theorem 2.2. Up to symmetry, all the solutions have been obtained. Here again, up to a scaling, the solutions w of types (1)–(4) are unique. δ ≤ min(α, N) (apart from α = δ = N) Theorem 4.2. Suppose ε = −1 and δ ≤ min(α, N ) (apart from α = δ = N ). Then all regular solutions of (E w ) have constant sign and a reduced domain (Sw < ∞). There exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →0 r α w = L 6= 0 and lim r →∞ r η w = c 6= 0 if δ < N , or (2–40) holds if δ = N < α; (2) w is positive, lim r →0 r α w = L 6= 0 if δ < α, or (2–39) holds if α = δ < N , and Sw < ∞; (3) w has one zero and the same behavior. Up to symmetry, all solutions are as above. Proof. Here (0, 0) is the only one stationary point, and N ≥ 2 (Figure 5). By Propositions 2.5 and 2.14, all regular solutions have constant sign, and Sw < ∞. Moreover w0 > 0 near 0, by Theorem 2.2; and w can only have minimal points, by Remark 2.3, so w 0 > 0 on (0, Sw ). In other words, Tr stays in Q4 , and limτ →ln Sw Y/y = −1. By Propositions 2.5 and 2.7, any solution y has at most one zero and is monotone at the extremities. By Proposition 2.8, apart from Tr , any

244


N 0; hence T starts from Q1 or Q3 at −∞. For any P = (ϕ, 0) with ϕ > 0, the trajectory T[P] goes from Q1 into Q4 at P, by Remark 2.1(i) on page 211; it stays in Q4 after P, since it cannot meet Tr ; and it stays in Q1 before P: it cannot start from Q3 , because it does not meet −Tr . Thus y remains positive and w is of type (2). For any P = (0, ξ ) with ξ > 0, T[P] goes from Q1 into Q2 by the same remark; thus T[P] stays in Q2 after P, since it cannot meet −Tr , and in Q1 before P, and w is of type (3). It remains to prove the existence of solutions of type (1). If δ < N , the origin is a saddle point, so there exists a trajectory T1 converging to (0, 0) at ∞; and limτ →∞ ζ = η > 0, by Proposition 2.8. Thus T1 lies in Q1 for large τ , and stays there, because Q1 is backward invariant. The conclusion follows. If δ = N , we consider the sets A = {P ∈ Q1 : T[P] ∩ {(ϕ, 0) : ϕ > 0} 6 = ∅}, B = {P ∈ Q1 : T[P] ∩ {(0, ξ ) : ξ > 0} 6 = ∅}.

They are nonempty and open, since the vector field is transverse at (ϕ, 0) and (0, ξ ); thus A ∪ B 6= Q1 . Hence there exists a trajectory T1 staying in Q1 ; therefore Sw = ∞ and T1 converges to (0, 0) at ∞, and w is of type (1), by Proposition 2.9. All solutions have been described, up to symmetry. 5. The case ε = 1, δ ≤ α N ≤δ≤α Theorem 5.1. Assume ε = 1, N ≤ δ ≤ α and α 6= N .


N =2 0, or limr →0 r α w = L 6= 0 if α 6= δ, or (2–39) holds if α = δ. Proof. (i) Here (0, 0) is the only stationary point. From Proposition 2.8, any trajectory is bounded and y is oscillating around 0 near ∞. First assume N < δ < α (Figure 6, left). Then (0, 0) is a source and all trajectories have a limit cycle at ∞ or are periodic. In particular there exists at least one cycle, with orbit O p . The trajectory Tr has a limit cycle O ⊆ O p . There exist also trajectories Ts starting from (0, 0) with an infinite slope, such that lim r →0 r η w = c 6= 0 if N ≥ 2 or (2–41) if N = 1, and all the Ts have the same limit cycle O. Next assume N = δ < α (Figure 6, right). Then Tr cannot converge to (0, 0), since it would intersect itself. Thus again the limit set at ∞ is a closed orbit O. No trajectory can converge to (0, 0) at ∞, as it would spiral around this point and hence intersect Tr . Consider any trajectory T 6= Tr in the connected component of O containing (0, 0). T is bounded, so its limit set at −∞ is (0, 0) or a closed orbit. The second case is impossible, since T does not meet Tr . Thus T is of the form Ts , and the corresponding w satisfies (2–40). (ii) By Theorem 2.21, all cycles are contained in a ball B of R2 . Take any point P0 exterior to B. Then T[P0 ] has a limit cycle at ∞ contained in B. If it has a limit cycle at −∞, it is contained in B, so T[P0 ] is contained in B, which is impossible. Thus y has constant sign near ln Rw . By Proposition 2.8, either Rw > 0 or y is defined near −∞.

246


Figure 7. Theorem 5.2: ε = 1, α = δ = N = 3. Theorem 5.2. Assume ε = 1 and α = δ = N . All regular solutions of (E w ) have constant sign, and are given by (1–6). For any k ∈ R, w(r ) = kr −N is a solution. There exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →0 r N w = c1 > 0, lim r →0 r N w = c2 > 0 (c2 6= c1 ); (2) w has one zero, lim r →0 r N w = c1 > 0 and lim r →∞ r N w = c2 < 0; (3) w is positive, Rw > 0, and lim r →0 r N w = c 6= 0; (4) w has one zero and the same behavior. Up to symmetry, all solutions are as above. Proof. Since α = N , equation (E w ) admits the first integral (1–5), so JN ≡ C for C ∈ R. We gave in (1–6) the regular (Barenblatt) solutions for the case C = 0. Since δ = N , (1–5) is equivalent to the equation Y ≡ y − C, by (2–12) (refer to Figure 7). For any k ∈ R, (y, Y ) ≡ (k, |N k| p−2 N k) is a solution of the system (S) located on the curve M, so that w(r ) = kr −N is a solution. Any solution has at most one zero, by Proposition 2.5. By Propositions 2.8 and 2.10, any trajectory converges to a point (k, |N k| p−2 N k) of M at ∞. Let C < 0 be such that the line Y = y − C is tangent to M. For any C ∈ (C, 0), the line Y = y − C cuts M at three points k1 < 0 < k2 < k3 . And y 0 > 0 if the trajectory is below M and y 0 < 0 if it is above M. We find two solutions y defined on R: one is positive and limτ →−∞ y = k2 , limτ →−∞ y = k3 , and the other has one zero. All other solutions satisfy Rw > 0, limτ →ln Rw Y/y = 1; some of them are positive, the others have one zero. δ < min(α, N) Here the system has three stationary points: (0, 0) is a saddle point, while M` , M`0 are sinks if δ ≤ N /2, or N /2 < δ and α < α ∗ , and sources when N /2 < δ and α > α ∗ , and node points whenever α ≤ α1 or α2 ≤ α, where α1 , α2 are defined in (2–48) (recall that α1 can be greater or less than η). This case is one


p -LAPLACE EQUATION

247

of the most delicate, since two types of periodic trajectories can appear, either surrounding (0, 0), corresponding to changing sign solutions, or located in Q1 or Q3 , corresponding to solutions of constant sign. Notice that δ < N implies δ < N < η by (1–2), and N /2 < δ implies η < α ∗ by (2–32). Remark 5.3. (i) Tr starts in Q1 . Since (0, 0) is a saddle point, Propositions 2.8 and 2.9 show there is a unique trajectory Ts converging to (0, 0), residing in Q1 for large τ , having an infinite slope at (0, 0), and satisfying lim r →0 r η w = c > 0. Moreover if Tr does not stay in Q1 , then Ts stays in it, and is bounded and contained in the domain delimited by Q1 ∩ Tr , by Remark 2.1(i). Thus if Tr is homoclinic, it stays in Q1 . (ii) Any trajectory T is bounded near ∞, by Propositions 2.8 and 2.12. From the strong form of the Poincaré–Bendixson theorem [Hubbard and West 1995, p. 239], any trajectory T bounded at ±∞ either converges to (0, 0) or ±M` , or its limit set 0± at ±∞ is a cycle, or it is homoclinic hence T = Tr and 0± = Tr (indeed, for any P ∈ 0± , T[P] converges at ∞ and −∞ to (0, 0) or ±M` ; if one of them is ±M` , then ±M` ∈ T[P] ⊂ 0± , and M` is a source or a sink, so T converges to ±M` ; otherwise T is homoclinic and T[P] = Tr ). (iii) If there exists a limit cycle around (0, 0), then by (2–42) this cycle also surrounds the points ±M` . We begin with the case α ≤ η, where there exists no cycle in Q1 and no homoclinic orbit, by Theorem 2.20. Theorem 5.4. Assume that ε = 1 and δ < min(α, N ), and α ≤ η. Then all regular solutions of (E w ) have constant sign, and lim r →∞ r δ |w(r )| = `. And w(r ) = `r −δ is a solution. If α < η, there exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →0 r α w = L and limr →∞ r δ w = `; (2) w is positive, Rw > 0 and lim r →∞ r η w = c > 0; (3) w is positive, Rw > 0 and lim r →∞ r δ w = `; (4) w has one zero, Rw > 0 and lim r →∞ r δ w = `; If α = η, then w = Cr −η is a solution and there exist solutions of type (4), but not of type (2) or (3). Proof. By Proposition 2.5 and Remark 2.3, Tr stays in Q1 and converges to M` at ∞; indeed there is no cycle in Q1 , by Propositions 2.8, 2.12 and 2.20. (i) Assume α < η (Figure 8, left). Consider any trajectory in Q1 . Then Yα > 0. If there exists τ such that Yα0 (τ ) = 0, at this point Yα00 (τ ) ≥ 0 by (2–36), and τ

248


α = 4.7 < η

α=5=η

Figure 8. Theorem 5.4: ε = 1, δ = 3 < N = 4 < η = 5. is a local minimum. Tr satisfies limτ →−∞ Yα = 0, and so Yα0 > 0 on R. This is equivalent to αy > Y 1/( p−1) + ( p − 1)(η − α)Y . Therefore Tr stays strictly below the curve Mα = (y, Y ) ∈ Q1 : αy = Y 1/( p−1) + ( p − 1)(η − α)Y . First consider Ts . Since α < η, this trajectory satisfies limτ →∞ Yα = 0. Then < 0 on (ln Rw , ∞), so Ts stays strictly above Mα . Hence it stays above M: indeed, if it meets M at a first point (y1 , (δy1 ) p−1 ), the function y has a maximum at this point. Thus by (2–16), we have ` < y1 and Yα0

2− p

(α − δ)y1

= δ p−1 ( p − 1)(η − α) < δ p−1 ( p − 1)(η − δ),

contradicting (1–2) and (1–4). This shows that y 0 < 0. Suppose that y is defined on R; then limτ →−∞ y = ∞ and limτ →−∞ ζ = α. If ζ 0 > 0 on R, then ζ (R) = (α, η), which contradicts (2–9). Then ζ has at least one extremal point τ , and ζ (τ ) is exterior to (α, η), by (2–9); if it is a minimum, ζ (τ ) > α by (2–18), since y 0 < 0, and if it is a maximum, ζ (τ ) < α. Thus we reach again a contradiction. Therefore Rw > 0 and limτ →ln Rw Y /y = 1, and the corresponding w is of type (2). For any P = (ϕ, 0), ϕ > 0, the trajectory T[P] stays in Q1 after P. The solution (y, Y ) originating at P at time 0 satisfies Yα (0) = 0; hence Yα0 (τ ) > 0 for any τ ≥ 0. Thus T[P] stays below Mα . Moreover it enters Q4 as τ decreases. But y 0 > 0 in Q4 , by (S); thus T[P] does not stay in Q4 , by Proposition 2.8; it goes into Q3 and must stay there because it cannot meet −Ts . This shows that Rw > 0 and y has precisely one zero, and w is of type (4). Next consider any trajectory T[P1 ] going through some point P1 = (y1 , Y1 ) in 1/( p−1) Q1 , lying below Ts and such that αy1 < Y1 . Such a trajectory exists because y = Y is an asymptotic direction of Ts . Let (y, Y ) be the solution issuing from P1 at time 0. Suppose y is defined on R; then limτ →−∞ y = ∞ and limτ →−∞ ζ = α.


p -LAPLACE EQUATION

249

Also, ζ (0) > α. Then ζ > δ on (−∞, 0): otherwise there would exist τ < 0 such that ζ (τ ) = α and ζ 0 (τ ) ≥ 0, contradicting (2–9). Thus y 0 < 0 on (−∞, τ1 ). Either ζ 0 > 0 on (−∞, 0), in which case ζ > η > 0 by (2–9), which is impossible; or ζ has at least an extremal point τ . If it is a minimum, then ζ (τ ) > α from (2–18); if it is a maximum, then ζ (τ ) < α; and again we reach a contradiction. Therefore Rw > 0, and the trajectory stays in Q1 and converges to M` , because there is no cycle in Q1 , by Theorem 2.20. Hence w is of type (3). Let O be the domain of Q1 bounded above by Ts . It is forward invariant. Any trajectory going through any point of O converges to M` at ∞. Either it meets the axis Y = 0 at some point (ξ, 0) with ξ > 0, or it stays in O, satisfies Rw > 0 and limτ →ln Rw T /y = 1, and meets Mα , since M` lies strictly below Mα . Let A = {P ∈ O : T[P] ∩ {(ϕ, 0) : ϕ > 0} 6 = ∅}, B = {P ∈ O : T[P] ∩ Mα 6 = ∅}.

These sets are nonempty and open: indeed, one can check that the intersection with Mα is transverse, because α 6 = η. Thus A ∪ B 6 = O, so there exists a trajectory T1 with w of type (1). (ii) Assume α = η (Figure 8, right). There is no positive solution with Rw > 0, thus no solution of type (2) or (3). Indeed all the trajectories stay below Ts , and Ts is defined by the equation ζ ≡ η, meaning that w ≡ Cr −η , or equivalently Yη ≡ C; thus Yη0 ≡ 0 and Ts = Mη . Consider any trajectory T[P] going through some point P = (ϕ, 0) with ϕ > 0, and the solution (y, Y ) issuing from P at time 0. Then Yη (0) = 0 and Yη < 0, so Yη0 = ηy − |Y |(2− p)/( p−1) Y > 0 on (−∞, 0), seeing that T[P] does not meet −Ts . Suppose Rw = 0. Then T[P] starts from Q3 , with limτ →−∞ ζ = α = η. Then limτ →−∞ yη = L < 0; thus limτ →−∞ Yη = −(α|L|)(2− p)/( p−1) . A straightforward computation gives 1 Yη00 = Yη0 N − |Y |(2− p)/( p−1) . p−1 This leads to Yη00 < 0 near −∞, which is impossible; thus Rw < ∞ and w is of type (4). Remarks. (i) For α ≤ η, both trajectories Tr and Ts stay in Q1 . (ii) When α ≤ N , one can verify that the regular positive solution y is increasing and y ≤ ` on R, so r δ w(r ) ≤ ` for any r ≥ 0. (iii) When α = N , we have Tr = {(ξ, ξ ) : ξ ∈ [0, `)}, and the corresponding solutions w are given by (1–6) with K > 0. And T3 = {(ξ, ξ ) : ξ > `)} is a trajectory corresponding to particular solutions w of type (3), given by (1–6) with K < 0.

250


Next we come to the most interesting case, where η < α. Lemma 5.5. Assume ε = 1, δ < min(α, N ) and η < α. If N /2 < δ and α < α ∗ and Ts stays in Q1 , Ts has a limit cycle at −∞ in Q1 or is homoclinic. If δ ≤ N /2, then Ts does not stay in Q1 . Proof. In any case M` is a sink, so Ts cannot converge to M` at −∞. Suppose Ts has no limit cycle in Q1 , and is not homoclinic and stays in Q1 . (This happens when δ ≤ N /2, by Proposition 2.11.) Then either limτ →−∞ y = ∞ and lim r →0 r α w = 3 6= 0, or Rw > 0. In either case, for any d ∈ (η, α), the function yd (τ ) = r d w = r d−δ y satisfies limτ →ln Rw yd = ∞ = limτ →∞ yd . Then it has a minimum point, contradicting (2–5). Theorem 5.6. Assume ε = 1 and N /2 < δ < min(α, N ). Then w(r ) = `r −δ is still a solution. (i) There exists a (maximal) critical value αcrit of α, such that max(η, α1 ) < αcrit < α ∗ , and the regular trajectory is homoclinic: all regular solutions of (E w ) have constant sign and satisfy lim r →∞ r η w = c 6= 0. (ii) For any α ∈ (αcrit , α ∗ ), there does exist a cycle in Q1 , in other words there exist positive solutions w such that r δ w is periodic in ln r . There exist positive solutions such that r δ w is asymptotically periodic in ln r near 0 and lim r →∞ r δ w = δ. There exist positive solutions such that r δ w is asymptotically periodic in ln r near 0 and lim r →∞ r η w = c 6= 0. (iii) For any α ≥ α ∗ there does not exist such a cycle, but there exist positive solutions such that lim r →0 r δ w = ` and lim r →∞ r η w = c > 0. (iv) For any α > αcrit , there exists also a cycle, surrounding (0, 0) and ±M` , thus r δ w is changing sign and periodic in ln r . All regular solutions change signs and are oscillating at ∞, and r δ w is asymptotically periodic in ln r . There exist solutions such that Rw > 0, or lim r →0 r α w = L 6= 0, and oscillating at ∞, and r δ w is asymptotically periodic in ln r . Proof. (i) For any α ∈ (α1 , α2 ) such that η ≤ α, we have by Remark 5.3 three possibilities for the regular trajectory Tr : •

Tr converges to M` and spirals around it, or else it has a limit cycle in Q1 around M` . Then Tr meets the set E = {(`, Y ) : Y > (δ`) p−1 } at a first point (`, Yr (α)). Note that ` and E depend continuously on α. Then Ts meets E at

some last point (`, Ys (α)) such that Ys (α) − Yr (α) > 0. See Figure 9, top left.


p -LAPLACE EQUATION

N < η = 5 < α = 5.1

N < η = 5 < α = 5.62

N < α = 5.9 < α ∗ = 6

N < α ∗ = 6 < α = 6.2

251

Figure 9. Theorem 5.6: ε = −1, N /2 < δ = 3 < N = 4.

•

•

Tr does not stay in Q1 ; then Ts is bounded at −∞, and so converges to M` at −∞ and spirals around this point, or it has a limit cycle around M` . Then Ts meets E at a last point (`, Ys (α)) and Tr meets E at a first point (`, Yr (α)) such that Ys (α) − Yr (α) < 0. See Figure 9, bottom row. Tr is homoclinic, which is equivalent to Ys (α) − Yr (α) = 0. See Figure 9, top

right. Now the function α 7→ g(α) = Ys (α) − Yr (α) is continuous. If α1 < η, then g(η) is defined and g(η) > 0, by Theorem 5.4. If η ≤ α1 , we observe that for α = α1 , the trajectory Ts leaves Q1 , by Theorem 2.18, because α1 is a sink, and does so transversally by Remark 2.1(i). The same holds for α = α1 +γ for γ small enough, by continuity, so Tr stays in Q1 and g(α1 + γ ) > 0. If α ≥ α ∗ (Figure 9, bottom right), then M` is a source or a weak source, by Theorem 2.16; thus Tr cannot converge to M` . By Theorem 2.19, there exists no cycle in Q1 and no homoclinic orbit. By Remark 5.3(i), Tr cannot stay in Q1 , so g(α) < 0 for α ∗ ≤ α < α2 . As a

252


consequence, there exists at least one αcrit ∈ (max(η, α1 ), α ∗ ) such that g(αcrit ) = 0. If it is not unique, we can choose the largest one. (ii) Suppose α < α ∗ . The existence and uniqueness of the desired cycle O in Q1 follows by Theorem 2.16 when α is close to α ∗ (Figure 9, bottom left). In fact, existence holds for any α ∈ (αcrit , α ∗ ); indeed g(α) < 0 on this interval, and Ts cannot converge to M` at −∞, so it has a limit cycle around M` at −∞. Since M` is a sink, there exist also trajectories converging to M` at ∞, with a limit cycle at −∞ contained in O. Now Tr does not stay in Q1 and is bounded at ∞, so it has a limit cycle at ∞ containing the three stationary points. (iii) Suppose α ≥ α ∗ . Then Ts stays in Q1 , is bounded on R, and converges at −∞ to M` . At the same time, Tr does not stay in Q1 for the same reason as above; thus it has a limit cycle at ∞, containing the three stationary points (see Figure 9, bottom right). (iv) For any α > αcrit , apart from Ts and the cycles, all the trajectories have a limit cycle at ∞ containing the three stationary points. By Theorem 2.21, all the cycles are contained in a ball B of R2 . Take any point P exterior to B. By Remark 5.3(ii), T[P] has a limit cycle at ∞ contained in B and cannot have a limit cycle at −∞. Thus y has constant sign near ln Rw . By Proposition 2.8, either Rw > 0 or y is defined near −∞ and limτ →−∞ ζ = L, lim r →0 r α w = L. Note. It is an open question whether αcrit is unique. It can be shown that if there 1 2 exist two critical values αcrit > αcrit , the first orbit is contained in the second. When δ ≤ N /2, or equivalently p ≤ P2 , there are no cycles in R2 and we get: Theorem 5.7. Assume ε = 1, δ ≤ N /2, and δ < α. All regular solutions of (E w ) have constant sign, and lim r →∞ r δ |w| = `. All solutions have a finite number of zeros. The function w(r ) = `r −δ is a solution. If α ≤ η, Theorem 5.4 applies. If η < α, all other solutions have at least one zero. There exist solutions satisfying lim r →∞ r η w = c 6= 0 and having m zeros, for some m > 0. All other solutions satisfy lim r →∞ r δ w = ±`, and have m or m + 1 zeros. There exist solutions with m + 1 zeros. Proof. (i) By Proposition 2.11, all solutions have a finite number of zeros. Since δ ≤ N /2, the function W defined in (2–21) is nonincreasing. The regular solutions (y, Y ) satisfy limτ →−∞ W (τ ) = 0, so W (τ ) ≤ 0 on R. If y(τ0 ) = 0 for some real 0 τ0 , then W (τ0 ) = |Y (τ0 )| p > 0, and we reach a contradiction. From Propositions 2.8 and 2.11 we obtain limτ →∞ y = ±`, so lim r →∞ r δ w = ±`. (ii) Assume η < α. By Lemma 5.5, Ts does not stay in Q1 . By Propositions 2.8 and 2.15, Ts cannot stay in Q4 , so it intersects the line y = 0 at points (0, ξ1 ), . . . , (0, ξm ). By Remark 5.3, any trajectory other than Ts converges to ±M` . Given


p -LAPLACE EQUATION

253

any P = (0, ξ ), with ξ > |ξi | for 1 ≤ i ≤ m, the trajectory T[P] cannot intersect Ts or −Ts , so y has m +1 zeros. Any other solution has m or m +1 zeros, because the trajectory does not meet Tr or −Tr or T[P] . Finally, Rw > 0 or lim r →0 r α w = L 6= 0. Note. Theorems 5.4, 5.6 and 5.7 recover, in particular, the results in [Qi and Wang 1999, Theorem 2]. 6. The case ε = −1, α ≤ δ max(α, N) ≤ δ Here (0, 0) is the only stationary point, and it is a source when δ 6= N . We first suppose 0 < α. Theorem 6.1. Suppose ε = −1, max(α, N ) ≤ δ and 0 < α. (i) Suppose α 6= N or α 6= δ. Then all regular solutions of (E w ) have constant sign and a reduced domain (Sw < ∞). There exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →0 r η w = c 6= 0 if N ≥ 2 (lim r →0 w = a > 0, lim r →0 w 0 = b < 0 if N = 1), and lim r →∞ r α w = L 6= 0 if α 6= δ, or (2–39) holds if α = δ; (2) w is positive, lim r →0 r η w = c 6= 0 if N ≥ 2 (lim r →0 w = a > 0, lim r →0 w 0 = b 6= 0, or a = 0 < b if N = 1), and Sw < ∞; (3) w has one zero, lim r →0 r η w = c 6= 0 if N ≥ 2 (lim r →0 w = a > 0, lim r →0 w 0 = b < 0 if N = 1), and Sw < ∞. (ii) Suppose α = δ = N . Then the regular solutions, given by (1–6), have constant sign, with Sw < ∞. For any k ∈ R, w(r ) = kr −N is a solution. Moreover there exist positive solutions such that lim r →0 r N w = c > 0 and Sw < ∞, and solutions with one zero, such that lim r →0 r N w = c > 0 and Sw < ∞. Up to symmetry, all solutions are as above. Proof. (i) Here α 6= N or α 6= δ (Figure 10, left). Since α > 0, Propositions 2.5, 2.7 and 2.14 imply that y > 0 and Sw < ∞ for Tr ; and any solution y has at most one zero, and y, Y are monotone near −∞ and near ln Sw . By Proposition 2.8, any trajectory T converges to (0, 0) at −∞; and apart from Tr , such a trajectory is tangent to the axis y = 0. Now suppose y > 0 near −∞. If N ≥ 2, then T starts in Q1 , since limτ →−∞ ζ = η > 0; if N = 1, then lim r →0 w = a ≥ 0 and lim r →0 w 0 = b, and T starts in Q1 if b < 0 and in Q4 if b > 0 (in particular when a = 0). For any P = (ϕ, 0) with ϕ > 0, the trajectory T[P] satisfies y > 0 on R, and by Remark 2.1(i), it stays in Q4 after P, because it cannot meet Tr (hence Sw < ∞); also it stays in Q1 before P, so w is of type (2). In the same way for any P = (0, ξ )

254


N = 2 < α = 2.5 < δ = 3

α=δ= N =3

Figure 10. Theorem 6.1: ε = −1. with ξ > 0, the trajectory T[P] stays in Q2 after P, since it cannot meet −Tr (hence Sw < ∞), and it stays in Q1 before P, so w is of type (3). Next consider the sets A = {P ∈ Q1 : T[P] ∩ {(ϕ, 0) : ϕ > 0} 6 = ∅}, B = {P ∈ Q1 : T[P] ∩ {(0, ξ ) : ξ > 0} 6 = ∅}.

From the previous discussion we know they are nonempty and open, so A∪ B 6= Q1 . There exists a trajectory T1 starting at (0, 0) and staying in Q1 . By Proposition 2.8, necessarily limτ →∞ y = ∞ and limτ →∞ ζ = α > 0, so w is of type (1) by Proposition 2.9. Finally we describe all other trajectories T[P] with one point P in the domain R above Tr ∪ (−Tr ). If P is in the domain delimited by Tr , T1 , then w is still of the type (2). If P is in the domain delimited by −Tr , T1 , then either y has a zero and w is of type (3), or N = 1, y < 0 and −w is of type (2). Up to a symmetry, all the solutions have been obtained. (ii) Here α = δ = N (Figure 10, right). Since α = N equation (1–5) holds, and the regular solutions relative to C = 0 are given by (1–6). Since δ = N , (1–5) is equivalent to y+Y ≡ C, from (2–12). For any k ∈ R, (y, Y ) ≡ Pk = (k, |N k| p−2 N k) is a solution of system (S), located on the curve M, thus w(r ) = kr −N is a solution of (E w ). Any solution has at most one zero, by Proposition 2.5. From Propositions 2.8, and 2.10, any other trajectory converges to a point Pk ∈ M at ∞, and Sw < ∞. There exists trajectories such that y has constant sign, and other ones such that y has one zero. All solutions have been obtained. Next we suppose α < 0, and distinguish the cases N ≥ 2 and N = 1. Theorem 6.2. Suppose ε = −1 and α < 0 < 2 ≤ N ≤ δ. Then any solution of (E w ) has a finite number of zeros. Regular solutions have at least one zero, and


p -LAPLACE EQUATION

255

precisely one if − p 0 ≤ α. Any solution has at least one zero, and any nonregular solution satisfies lim r →0 r η w = c 6= 0. If − p 0 < α, all regular solutions have a reduced domain (Sw < ∞), and they fall into the following types, all of which occur: (1) solutions with two zeros and Sw < ∞; (2) solutions with one zero and lim r →∞ r α w = L 6= 0; (3) solutions with one zero and Sw < ∞. If α = − p 0 , all regular solutions satisfy lim r →∞ r α w = L 6= 0. The other solutions are of type (1). Proof. By Proposition 2.8, any trajectory converges necessarily to (0, 0) at −∞, and apart from Tr , it is tangent to the axis y = 0. Any solution y has a finite number of zeros, and y is monotone near −∞, and near Sw (finite or not), by Propositions 2.7 and 2.11, since δ > N /2. Either Sw < ∞, so limτ →ln Sw Y/y = −1, or Sw = ∞ and limτ →∞ ζ = α < 0. In any case (y, Y ) is in Q2 or Q4 for large τ . By Proposition 2.14, Tr has at least one zero, and starts in Q1 . Since N ≥ 2, any trajectory T 6= ±Tr satisfies limτ →−∞ ζ = η > 0. Thus it starts in Q1 (or Q3 ), and has at least one zero. Any trajectory T starting in Q1 enters Q2 , by Remark 2.1(i). And y 0 = δy − Y 1/( p−1) , so y decreases as long as T stays in Q2 . Then either T enters Q3 , hence also Q4 , and y has at least two zeros; or it stays in Q2 , and either Sw < ∞ and limτ →ln Sw Y/y = −1, or Sw = ∞ and limτ →∞ ζ = α. (i) Suppose − p 0 < α (Figure 11, left). Then Tr has precisely one zero, by Proposition 2.14, thus it stays in Q2 , and Sw < ∞, limτ →ln Sw Y/y = −1. Any other solution has at most two zeros, because the trajectory does not meet ±Tr . Recall that the function Yα defined by (2–3) with d = α has only minimal points on the

− p 0 = −3 < α = −2.5 < 0 < N

− p 0 = −3 = α < 0 < N

Figure 11. Theorem 6.2: ε = −1, N = 2 < δ = 3.

256


sets where it is positive, by Remark 2.6. By Proposition 2.14, Tr satisfies Yα0 = −( p − 1)(η − α)Yα + e( p−(2− p)α)τ (Yα1/( p−1) − αyα ) > 0, which is equivalent to (6–1)

Y 1/( p−1) − ( p − 1)(η − α)Y > αy.

Tr stays strictly to the right of the curve

(6–2)

Nα = {(y, Y ) ∈ R × (0, ∞) : αy = Y 1/( p−1) − ( p − 1)(η − α)Y },

which intersects the axis y = 0 at the points (0, 0) and (0, ( p − 1)(η − α)). For P = (ϕ, 0) with ϕ < 0, the trajectory T[P] enters Q3 after P, by Remark 2.1(i); the solution passing through P at τ = 0 satisfies Yα (0) = 0 (so Yα stays positive for τ < 0) and Yα0 (τ ) < 0, since Yα has no maximal point. Thus T[P] stays in Q1 ∪ Q2 before P, to the left of Nα , and starts and (0, 0) in Q1 and ends up in Q4 . Hence y has two zeros. If Sw = ∞ then limτ →∞ |y| = ∞ and limτ →∞ ζ = α < 0; this is impossible, because T[P] does not meet −Tr . Thus Sw < ∞, and w is of type (1). Next consider T[P] , for P = (ϕ, ξ ) ∈ Nα , with ϕ ≤ 0. The solution going through 00 P at τ = 0 satisfies Yα0 (0) = 0, Yα (0) > 0, and 0 is a minimal point; hence Yα (0) > 0. Indeed, if Yα00 (0) = 0, then Yα is constant on R by uniqueness; by (2–6), in turn, we have Yα ≡ 0 (since α 6= − p 0 ); but this is false. Therefore Yα0 (τ ) > 0 for τ > 0 and Yα0 (τ ) < 0 for τ < 0. Thus T[P] stays in Q1 ∪ Q2 , to the right of Nα after P, with y < 0 by Remark 2.1(i); it stays to the left of Nα before P, and converges to (0, 0) at −∞ in Q1 . Suppose that Sw = ∞. Then limτ →∞ |y| = ∞, limτ →∞ ζ = α, and limτ →∞ yα = L < 0 by Proposition 2.9; thus limτ →∞ Yα = (αL) p−1 . As in Proposition 2.14, one finds that Yα00 (τ ) > 0 for any τ > 0, which is impossible. Thus T[P] satisfies Sw < ∞, showing that limτ →ln Sw Y/y = −1. The corresponding w is of type (3). Finally, let R be the domain of Q1 ∪ Q2 delimited by Tr and containing Nα , and define the sets (6–3)

A = {P ∈ R : T[P] ∩ {(ϕ, 0) : ϕ < 0} 6 = ∅}, B = {P ∈ R : T[P] ∩ Nα 6 = ∅},

corresponding to trajectories of type (1) or (3). These sets are nonempty and open, because here again the intersection with Nα is transverse (recall that α 6= − p 0 ). Thus A ∪ B 6= R. There exists a trajectory in R disjoint from Nα , starting at (0, 0) in Q1 and ending up in Q2 . It cannot satisfy limτ →ln Sw Y/y = −1, so Sw = ∞ and limτ →∞ ζ = α. Hence w is of type (2).


p -LAPLACE EQUATION

257

(ii) Suppose α = − p 0 (Figure 11, right). The regular solutions are given by (1–8), they have one zero, but Sw = ∞ and limτ →∞ ζ = α. They satisfy Y− p0 ≡ C, thus Y−0 p0 ≡ 0, thus Tr = M− p0 . Consider T[P] ; the solution passing through P at τ = 0 satisfies and Y− p0 (0) = 0, thus Y− p0 stays negative for τ > 0 and Y−0 p0 < 0. Suppose that Sw = ∞, then limτ →∞ yα = L > 0, limτ →∞ Yα = −(|α|L) p−1 . But as in (2–46), Yα00 (τ ) < 0 for any τ > 0, which leads to a contradiction. Thus Sw < ∞, and w is of type (1). Finally suppose that there exists a trajectory T 6= Tr staying in Q1 ∪ Q2 . Then Yα > 0, limτ →∞ Yα = 0, and it cannot meet Tr , thus Sw = ∞, and limτ →−∞ Yα = ∞, limτ →∞ Yα = C > 0. As in Proposition 2.14, it is impossible. Thus there does not exist solution of type (2) or (3). Theorem 6.3. Suppose ε = −1 and α < 0 < N = 1 < δ. Then any solution of (E w ) has still a finite number of zeros. Regular solutions have at least one zero, and precisely one if − p 0 ≤ α. If −1 < α < 0, all regular solutions have a reduced domain (Sw < ∞). Moreover: (1) the solutions with lim r →0 w = a > 0 and lim r →0 w 0 = b < 0 have one zero and Sw < ∞; (2) the solutions with lim r →0 w = 0 and lim r →0 w0 = b > 0 are positive and Sw < ∞; (3) there exist solutions with one zero and lim r →0 w = a > 0, lim r →0 w0 = b > 0 and Sw < ∞; (4) there exist positive solutions with lim r →0 w = a > 0, limr →0 w 0 = b > 0 and Sw < ∞; (5) for any a > 0 there exists b > 0 such that w is positive and lim r →∞ r α w = L 6= 0. If α = −1, for any b > 0, w ≡ br is a solution. The other solutions such that lim r →0 w 6= 0 have one zero, and satisfy Sw < ∞. If − p 0 < α < −1, then (6) there exist solutions with one zero, with lim r →0 w = a > 0, lim r →0 w0 = b < 0, and Sw < ∞; (7) the solutions with lim r →0 w = 0 and lim r →0 w 0 = b > 0 have one zero and Sw < ∞; (8) there exist solutions with one zero, with lim r →0 w = a > 0, lim r →0 w0 = b > 0 and Sw < ∞; (9) there exist solutions with lim r →0 w = a > 0, lim r →0 w0 = b < 0, with two zeros and Sw < ∞;

258


−1 < α = −0.5 < N

− p 0 = −3 < α = −2.9 < −1 < N

Figure 12. Theorem 6.3: ε = −1, N = 1 < δ = 3. (10) for any a > 0 there exists b > 0 and a solution with lim r →0 w = a > 0, lim r →0 w0 = b < 0, with one zero and lim r →∞ r α w = L 6= 0. Proof. The case N = 1 is still the more complex one, since some trajectories start in Q2 (or Q4 ), corresponding to the solutions such that lim r →0 w = a and lim r →0 w0 = b, with b 6= 0, ab ≥ 0. Any solution has still a finite number of zeros, by Proposition 2.11. (i) Suppose −1 < α < 0 (Figure 12, left). By Proposition 2.5, any solution has at most one zero, so regular solutions have precisely one zero. Thus Tr meets the axis y = 0 at some point (0, ξr ). Consider the trajectory Ts such that limr →0 w = 0 and lim r →0 w 0 = b < 0 (which means limτ →−∞ ζ = η = −1), starting from (0, 0) in Q2 , so w < 0 near 0. For any d ∈ (−1, α), the function yd satisfies yd (τ ) = be(d+1)τ (1 + o(1)) near −∞, so limτ →−∞ yd = 0. Then yd has no zeros, because |yd | has no maximal point, by (2–14); thus Ts stays in Q2 . If Ts satisfies Sw = ∞, then limτ →∞ yα = L < 0, so limτ →∞ yd = 0, which is impossible; thus w is of type (2). The domain is reduced since Tr cannot meet Ts . For P = (ϕ, 0) with ϕ < 0, the trajectory T[P] does not meet Ts , thus converges to (0, 0) at − ∞ in Q2 ; then limr →0 (−w) = a > 0 and lim r →0 (−w)0 = b > 0, and T[P] ends up in Q4 ; thus y has one zero and −w is of type (3). For P = (0, ξ ), with ξ ∈ (0, ξr ), T[P] has one zero and converges to (0, 0) at −∞ in Q1 ; hence lim r →0 w = a > 0 and lim r →0 w0 = b < 0. The domain is reduced since T[P] and Ts do not meet. Thus w is of type (1). Conversely, any solution such that lim r →0 w = a > 0 and lim r →0 w0 = b < 0 has one zero and satisfies Sw < ∞. Next consider a trajectory T such that limr →0 (−w) = a > 0 and lim r →0 (−w0 ) = b > 0; that is, T starts in Q2 below Ts . Then ζ (τ ) = −(b/a)eτ (1 + o(1) near −∞,


p -LAPLACE EQUATION

259

so limτ →−∞ ζ = 0. If ζ has an extremal point θ , we have ( p − 1)ζ 00 (θ) = (2 − p)(ζ − α)(δ − ζ )|ζ y|2− p , by (2–18); thus θ is a minimal point if ζ (θ) > α, and maximal if ζ (θ) < α. (Equality is impossible since it would require ζ ≡ α.) Thus either ζ has a first zero τ1 and α < ζ (τ ) < 0 for τ < τ1 , and T is one of the T[P] ; or ζ remains negative, in which case if Sw = ∞, then limτ →∞ ζ = α, so ζ is necessarily decreasing, and α < ζ (τ ) < 0 for any τ . In both cases, T stays below the curve M0 = (y, Y ) ∈ R × (0, ∞) : αy = Y 1/( p−1) , as long as it is in Q2 . Hence, for any P ∈ Q2 such that P is on or above M0 , the trajectory T[P] satisfies Sw < ∞; in particular on finds again Ts . For any P between M0 and Ts , the solution has constant sign, T[P] converges to (0, 0) at −∞ and lim r →0 (−w) = a > 0 and lim r →0 (−w0 ) = b > 0, and limτ →ln Sw Y/y = −1, so T[P] meets Mα . Thus −w is of type (4). Finally, let R1 be the domain of Q2 delimited by Ts and the axis Y = 0, and set A1 = P ∈ R1 : T[P] ∩ {(ϕ, 0) : ϕ < 0} 6 = ∅ , B1 = P ∈ R1 : T[P] ∩ Nα 6 = ∅ . These sets are open, since the intersection is transverse (recall that α 6= −1). They are also nonempty, so A1 ∪ B1 6= R1 , and there exists a trajectory such that y is defined on R and limτ →∞ ζ = α. By scaling, we can find for any a > 0 at least one b such that the corresponding w has constant sign and lim r →∞ r α w = L 6= 0; thus |w| is of type (5). (ii) Suppose α = −1. Then Ts is given explicitly by w ≡ br , so Y ≡ −y p−1 , or 00 equivalently Y−1 ≡ b; hence Ts = N−1 . For any other solution, one finds Y−1 = 0 00 2τ (2− p)/( p−1) Y−1 (1 + e |Y−1 | ), so Y−1 is strictly monotone, by uniqueness, and Y−1 0 has the sign of Y−1 . Any trajectory such that lim r →0 w = a > 0 and lim r →0 w0 = 0 b < 0, starting in Q1 , satisfies Y−1 > 0, and Y−1 is convex. Thus Y−1 cannot have a finite limit, Sw < ∞, and the trajectory ends up in Q2 , so y has a zero. Any trajectory such that lim r →0 (−w) = a > 0 and lim r →0 (−w)0 = b > 0, starting in 0 Q2 , satisfies Y−1 < 0, so Y−1 has a zero and the trajectory ends up in Q4 . Hence, apart from Ts , all trajectories satisfy Sw < ∞, and y has one zero. (iii) Suppose − p 0 < α < −1 (Figure 12, right). Then Tr starts in Q1 , y has one zero from Proposition 2.14, and Tr ends up in Q2 , with Sw < ∞. Any solution has at most two zeros. Consider Ts : we claim that it cannot stay in Q2 . Suppose that it stays in it, thus y < 0 < Y . Then ζ < 0, and limτ →−∞ ζ = η = −1, and ζ is monotone near −∞; if ζ 0 ≤ 0, then ζ ≤ −1 near −∞, and we reach a contradiction from (2–9). Then

260


ζ 0 ≥ 0 near −∞; but any extremal point of ζ is a minimal point by (2–18). Hence ζ remains increasing, is defined on R and has a limit λ ∈ [−1, 0]; but λ = α, by Proposition 2.8, again leading to a contradiction. Therefore Ts enters Q3 at some point (ϕs , 0) with ϕs < 0, then enters Q4 , and y has precisely one zero; and w is of type (7). Any solution such that lim r →0 (−w) = a > 0 and lim r →0 (−w)0 = b > 0 also has one zero, since its trajectory stays under Ts in Q2 ; thus w is of type (8). As in the case N ≥ 2, for any P = (ϕ, ξ ) ∈ Nα with ϕ ≤ 0, T[P] stays in Q1 ∪ Q2 and Sw < ∞. In particular for P0 = (0, ξ0 ), where ξ0 = (( p−1)(−1−α))( p−1)/(2− p) , the trajectory T[P0 ] starts from Q1 , so lim r →0 w = a > 0, lim r →0 w0 = b0 (a) > 0; also w has one zero, and Sw < ∞. Thus w is of type (6). The sets A, B defined as in (6–3) are still open in this case, and B contains T[P0 ] . Also, A contains Ts ; hence A contains any T[P] , where P = (ϕ, 0) with ϕ < ϕs . Such a trajectory satisfies lim r →0 w = a > 0 and lim r →0 w0 = b < 0, and w is of type (9). Moreover A ∪ B 6= R; thus for any a > 0 there exists b < 0 such that the corresponding w has one zero and lim r →∞ r α w = L 6= 0, so w is of type (10). α 0; (4) w is positive, lim r →0 r δ w = ` and lim r →∞ r α w = L > 0. Up to symmetry, all solutions are as above. Proof. Since α > 0, regular solutions have constant sign and satisfy Sw < ∞, by Propositions 2.5 and 2.14. Here Tr starts in Q4 and stays in it, by Remark 2.3 (Figure 13). Any solution has at most one zero by Proposition 2.5. The point M` is a source, and a node point, by Remark 2.17, and 0 < λ1 < δ < λ2 . The eigenvectors u 1 = (ν(α), λ1 −δ) and u 2 = (−ν(α), δ−λ2 ) form a positively oriented basis, where


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now ν(α) < 0; thus u 1 points toward Q3 and u 2 toward Q4 . There are two particular trajectories T1 , T2 starting from M` at −∞, with respective tangent vectors u 2 and −u 2 . All other trajectories T approaching M` at −∞ do so along u 1 ; and y is monotone at the extremities, by Proposition 2.7, since T cannot meet T1 , T2 . First consider T1 . The function y is nondecreasing near −∞ and remains so as long as T1 stays in Q1 ; indeed, Y is nonincreasing near −∞, so Y (τ ) < (δ`) p−1 . If y has a maximal point τ , then y(τ ) > ` by (2–16), and Y 1/( p−1) = δy; hence Y (τ ) > (δ`) p−1 , so Y has a minimal point τ1 in Q1 ; therefore Y (τ1 ) < (δ`) p−1 by (E Y ); and Y 0 (τ1 ) = 0, so α` < αy(τ1 ) < (N − δ)αY (τ1 )/(δ − α), a contradiction. If T1 stays in Q1 , then limτ →−∞ ζ = α > 0 by Proposition 2.8, which is also contradictory. Thus T1 enters Q4 at some point (ϕ1 , 0) and stays in it; Sw < ∞ because T1 and Tr don’t meet, so w is of type (1). Next consider T2 . Near −∞, the function Y is nondecreasing, and y is nonincreasing; y is monotone as long as y > 0: if there existed a minimal point τ , we would have y(τ ) > ` by (2–16). Also Y is nondecreasing as long as Y > 0: if Y has a maximal point τ , then Y (τ ) > (δ`) p−1 by (E Y ); and α` > αy(τ ) > (N − δ)αY (τ )/(δ − α), which is again impossible. Thus T2 cannot stay in Q1 ; it enters Q2 at some point (0, ξ2 ) and stays in Q2 , since it does not meet −Tr . Hence Sw < ∞, and w is of type (2). There exists also a unique trajectory T3 converging to (0, 0) at ∞, ending up in Q1 , since (0, 0) is a saddle point. It stays in the domain of Q1 delimited by T1 , T2 , because Q1 is backward invariant. Thus T3 converges to M` at −∞, tangentially to u 1 . And y is increasing on R: indeed y 0 < 0 near ±∞, and y cannot have two extremal points. Then w is of type (3). For any point P = (ϕ, 0) with ϕ > ϕ1 , the trajectory T[P] goes from Q1 into Q4 , by Remark 2.1(i). It does not meet Tr or T1 ; hence it stays in Q4 after P, and

Figure 13. Theorem 6.4: ε = −1, 0 < α = 2 < δ = 3 < N = 4.

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Sw < ∞. Before P, it stays in Q1 because it does not meet T1 or T2 , by the same remark. By Proposition 2.8, either limτ →−∞ ζ = α < δ, so y 0 = y(δ − ζ ) > 0 near −∞, and limτ →−∞ y = ∞, which is impossible; or (necessarily) T[P] converges to M` , tangentially to u 1 , and T[P] is of type (2). Similarly, for any P 0 = (0, ξ ) with ξ > ξ2 , the trajectory T[P 0 ] goes from Q1 into Q2 ; it remains there after P (so Sw < ∞) and remains in Q1 before P, converging to M` at −∞, tangentially to −u 1 . Thus w is still of type (2). The sets A = {P ∈ Q1 : T[P] ∩ {(ϕ, 0) : ϕ > 0} 6 = ∅}, B = {P ∈ Q1 : T[P] ∩ {(0, ξ ) : ξ > 0} 6 = ∅},

are open and nonempty, so A ∪ B 6= Q1 . There is at least one trajectory T4 in Q1 converging to M` at −∞ and such that limτ →∞ ζ =α; thus w is of type (4). For any point P in the bounded domain R0 of Q1 delimited by T2 and T3 , the trajectory T[P] is confined to R0 before P, and y has no maximal point; thus y is monotone, and T converges to M` at −∞. It cannot stay in Q1 since it cannot converge to (0, 0). Thus it goes from Q1 into Q2 and stays there, because it does not meet −Tr . Thus Sw < ∞, and w is again of type (2). For any P in the domain of Q1 delimited by T1 and T3 , the trajectory T[P] converges to M` at −∞, tangentially to u 1 ; it enters Q4 and stays there. Thus Sw < ∞ and w is of type (1). No trajectory can stay in Q4 (Q2 ) except Tr (−Tr ); thus all the solutions have been described, up to a symmetry. Now we come to the case α < 0, and discuss according to the sign of α− p 0 . This situation is different from the case ε = 1, δ < min(α, N ) discussed on page 246, by Remark (i) on page 249 and part (i) of the next remark. Remark 6.5. Assume ε = −1 and α < 0. (i) The regular trajectory Tr starts in Q1 . There exists a unique trajectory Ts converging to (0, 0), lying in Q1 for large τ , having an infinite slope at (0, 0), and satisfying lim r →0 r η w = c > 0. If Ts does not stay in Q1 , then Tr does stay in it, and it is bounded and contained in the domain delimited by Q1 ∩ Ts , by Remark 2.1(i). If Tr is homoclinic, it stays in Q1 . Conversely, if Ts stays in Q1 and is not homoclinic, Tr does not stay in Q1 , for the following reason. Ts either converges to M` at −∞ or has a limit cycle around it; if Tr stays in Q1 , either the corresponding y is increasing, so limτ →ln Sw Y /y = −1; or limτ →∞ ζ = α < 0, by Propositions 2.15 and 2.8, so Tr enters Q4 and we reach a contradiction; or y oscillates around ` near ∞, by Proposition 2.7, so it meets Ts , which is impossible. (ii) Any trajectory T is bounded near −∞ from Propositions 2.8 and 2.10. Any trajectory T bounded at ±∞ converges to (0, 0) or ±M` , or its limit set 0± at ±∞ is a limit cycle; or Tr is homoclinic and 0± = Tr .


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(iii) If there exists a limit cycle around (0, 0), it also surrounds ±M` , by (2–42) and (2–43). Next we study the case − p 0 ≤ α, where there is no cycle and no homoclinic orbit in Q1 , by Theorem 2.20. Theorem 6.6. (i) Assume ε = −1 and − p 0 < α < 0 < δ < N . Then all regular solutions have precisely one zero, and Sw < ∞. The function w ≡ `r −δ is a solution. There exist solutions satisfying any one of these characterizations: (1) w is positive, lim r →0 r δ w = ` and limr →∞ r η w = c > 0; (2) w has one zero, lim r →0 r δ w = `, and lim r →α r α w = L < 0; (3) w has one zero, lim r →0 r δ w = `, and Sw < ∞; (4) w has two zeros, lim r →0 r δ w = `, and Sw < ∞. (ii) Assume α = − p 0 . Then the regular solutions, given by (1–8), have one zero, and lim r →α r α w = L < 0. There exist solutions of type (1) and (4). Up to symmetry, all solutions are as above. Proof. (i) Assume − p 0 < α < 0 (Figure 14, left). By Proposition 2.5, any solution y has at most two zeros, and Y has at most one zero. First consider Ts . The function Yα defined by (2–3) with d = α satisfies Yα = O(e(α−η)τ ) near ∞, thus limτ →∞ Yα = 0. Then from Remark 2.6, Yα is decreasing, thus Yα > 0, and Ts stays in Q1 ∪ Q2 . In fact it stays in Q1 , by Remark 2.1(i). From Propositions 2.8, 2.7, 2.11, and Theorem 2.20, Ts converges to M` at −∞. Indeed if lim y = ∞, then limτ →∞ ζ = α < 0; if Sw < ∞, then lim Y/y = −1; which contradicts Y > 0. Then w is of type (1). The trajectory Tr stays in Q1 ∪ Q2 , and y has precisely one zero, and Sw < ∞, so limτ →ln Sw Y /y = −1. We claim that Tr cannot stay in Q1 . Indeed, it cannot

− p 0 = −3 < α = −2 < δ

− p 0 = −3 = α < δ

Figure 14. Theorem 6.6: ε = −1, δ = 3 < N /2 < N = 9.

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converge to M` , which is a source, or oscillate around Q1 , because it does not meet Ts , or tend to ∞, or satisfy Sw < ∞ with Y > 0. Thus y has precisely one zero, Tr enters Q2 and stays in it. Moreover the corresponding Yα satisfies Yα0 > 0, or equivalently (6–1). Consider again the curve Nα defined in (6–2). Here Tr stays strictly to the right of Nα , and Ts to the left of Nα . For any P = (ϕ, 0) with ϕ < 0, the trajectory T[P] enters Q3 after P, by Remark 2.1(i). The solution going through P at τ = 0 satisfies Yα (0) = 0; thus Yα stays positive as before, and Yα0 < 0, since Yα has no maximal point, by Remark 2.6. Thus T[P] stays in Q1 ∪ Q2 before P, to the left of Nα . It cannot stay in Q2 , by Propositions 2.7 and 2.8. As τ decreases, it enters Q1 , and converges to M` , by Theorem 2.20. If Sw = ∞, then lim |y| = ∞ and limτ →∞ ζ = α < 0; this is impossible, since T[P] does not meet −Tr . Thus Sw < ∞, lim Y/y = −1, T[P] goes from Q3 into Q4 and stays in it, and w is of type (4). The solution y has precisely two zeros. Next consider T[P] for any P = (ϕ, ξ ) ∈ Nα with ϕ < 0. The solution passing through P at τ = 0 satisfies Yα0 (0) = 0 and Yα (0) > 0, and 0 is a minimal point. Therefore Yα00 (0) > 0; indeed, if Yα00 (τ ) = 0, we conclude from uniqueness that Yα is constant on R; then (2–6) yields Yα ≡ 0, since α 6= − p 0 . But this cannot be. Therefore Yα0 (τ ) > 0 for τ > 0, Yα0 (τ ) < 0 for τ < 0, and T[P] stays in Q1 ∪ Q2 , to the right of Nα after P, with y < 0 by Remark 2.1(i), and to the left of Nα before P. As above it cannot stay in Q2 near −∞, and converges to M` . Suppose that it satisfies Sw = ∞. Then lim |y| = ∞, limτ →∞ ζ = α, and limτ →∞ yα = L < 0 by Proposition 2.9; hence limτ →∞ Yα = (αL) p−1 . As in Proposition 2.5(iii), we find Yα00 (τ ) > 0 for any τ > 0, which is impossible. Then Sw < ∞, so limτ →ln Sw Y/y = −1 and w is of type (3). Finally consider the domain R of Q1 ∪ Q2 delimited by Tr and Ts and containing Nα . Form the sets A = {P ∈ R : T[P] ∩ Nα 6 = ∅}, B = {P ∈ R : T[P] ∩ {(ξ, 0) : ξ > 0} 6 = ∅},

corresponding to trajectories of type (3) or (4). They are nonempty and open, since here again the intersection with Nα is transverse (α 6= − p 0 ). Thus A ∪ B is distinct from R: there exists a trajectory in R that does not meet Nα ; it converges to M` at −∞ or oscillates around it, and it is located below Nα in Q2 . It cannot satisfy limτ →ln Sw Y /y = −1, so Sw = ∞ and we have limτ →∞ ζ = α. Hence w is of type (2). (ii) Assume α = − p 0 (Figure 14, right). Then regular solutions have a different behavior: they are given explicitly by (1–8). They satisfy Y− p0 ≡ C, thus Y−0 p0 ≡ 0,


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thus Tr = M− p0 . Here y has a zero, and Sw = ∞, and limτ →∞ ζ = − p 0 . As above Ts stays in Q1 and w is of type (1). Next consider again T[P] . The solution going through P at τ = 0 satisfies Y− p0 (0) = 0, thus Y− p0 stays negative for τ > 0 and Y−0 p0 < 0. Suppose that Sw = ∞, and limτ →∞ ζ = − p 0 , then limτ →∞ yα = L > 0, limτ →∞ Yα = −(|α|L) p−1 . But as in (2–46), Yα00 (τ ) < 0 for any τ > 0, which leads to a contradiction. Then Sw < ∞ and w is of type (4). Finally suppose that there exists a trajectory T 6= Tr staying in Q1 ∪ Q2 . Then it converges to M` , thus Yα > 0, Sw = ∞, and limτ →−∞ Yα = ∞, limτ →∞ Yα = C > 0. If T has a minimal point, then it has an inflection point where Yα0 > 0, which as above is impossible. Then Yα0 < 0; (2–6) yields 0 (2− p)/( p−1) − N ( p − 1) = Y−0 p0 (Y − N ( p − 1)), ( p − 1)Y−00 p0 = Y−0 p0 e p τ Y− p0 and limτ →∞ Y = ∞, so Y−00 p0 < 0 for large τ , which is impossible. Thus there exist no solutions of type (2) or (3). We now come to the most difficult case: α < − p 0 . Lemma 6.7. Assume ε = −1 and α < − p 0 . If δ < N /2 and α ∗ < α, either Tr has a limit cycle in Q1 , or is homoclinic, or all regular solutions have at least two zeros. If N /2 ≤ δ < N , they have at least two zeros. Proof. In any case M` is a source. Suppose that Tr has no limit cycle in Q1 , or is not homoclinic (in particular it happens when N /2 ≤ δ < N , by Proposition 2.11), and stays in Q1 ∪ Q2 , thus Y stays positive. Then from Propositions 2.8, 2.9 and 2.15, either limτ →−∞ y = ∞, limτ →∞ yα = L 6= 0, limτ →∞ Yα = (αL) p−1 , or Sw < ∞. In any case, for any d ∈ (α, − p 0 ), the function Yd = e(d−α)τ Yα satisfies limτ →ln Sw Yd = ∞ = limτ →∞ Yd . Then it has a minimum point, and this contradicts (2–15). Thus Tr enters Q3 . If it stays in it, it has a limit cycle; then −Tr has a limit cycle in Q1 . But −Tr does not meet Tr , and M` is in the domain of Q1 delimited by Tr , since Tr meets M to the right of M` , by (2–16); this is impossible. Then Tr enters Q4 , and y has at least two zeros. Theorem 6.8. Assume ε = −1 and δ < N /2, α < − p 0 . Then w(r ) = `r −δ is still a solution. (i) There exists a (minimal) critical value α crit of α, such that α ∗ < α crit < min(− p 0 , α2 ) < 0, and Tr is homoclinic: all regular solutions have constant sign and satisfy lim r η w = c 6= 0.

r →∞

266


α = −5 < − p 0 = −3 < 0 < δ

α = −7.4 < 0 < δ

α ∗ = −9 < α = −8 < δ

α = −13 < α ∗ = −9 < δ

Figure 15. Theorem 6.8: ε = −1, δ = 3 < N /2 < N = 9. (ii) For any α ∈ (α ∗ , α crit ) there does exist a cycle in Q1 ; equivalently there exist solutions such that r δ w is periodic in ln r . All regular solutions have constant sign and r δ w is asymptotically periodic in ln r . There exist positive solutions such that lim r →0 r δ w = ` and r δ w is asymptotically periodic in ln r . (iii) For any α ≤ α ∗ , there does not exist such a cycle, regular solutions have constant sign, and lim r →∞ r δ |w| = `. (iv) For any α < α crit , there exists also a cycle surrounding (0, 0) and ±M` , thus w is changing sign and r δ w is periodic in ln r . There exist solutions oscillating near 0, and r δ w is asymptotically periodic in ln r , and lim r →∞ r η w = c 6= 0. There exist solutions oscillating near 0, and r δ w is asymptotically periodic in ln r , and Sw < ∞ or lim r →∞ r α w = L 6= 0. Proof. (i) For any α ∈ (α1 , α2 ), such that α ≤ − p 0 we have three possibilities, by Remark 6.5: •

Ts converges to M` at −∞, spiraling around this point, since α is a spiral point, or it has a limit cycle around M` . Then Ts meets the set E = {(`, Y ) :


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Y > (δ`) p−1 } at a first point (`, Ys (α)); and Tr meets E at a last point (`, Yr (α)) such that Yr (α) − Ys (α) > 0. Moreover Tr enters Q2 , by Proposition 2.8. See Figure 15, top left. •

Ts enters Q4 ; hence Tr converges to M` at ∞ and spirals around this point, or it has a limit cycle around M` . Then Ts meets E at a last point (`, Ys (α)), Tr meets E at a first point (`, Yr (α)) such that Yr (α) − Ys (α) < 0. See Figure

15, bottom row. •

Tr is homoclinic, or equivalently Yr (α) − Ys (α) = 0. See Figure 15, top right.

Now the function α 7→ h(α) = Yr (α) − Ys (α) is continuous. If − p 0 < α2 , then h(− p 0 ) is defined and h(− p 0 ) > 0, by Theorem 6.6. If α2 ≤ − p 0 , we observe that for α = α2 , by Theorem 2.18, Tr must leave Q1 (because α2 is a source) and does so transversally; thus the same holds for α = α2 − γ if γ > 0 is small enough. Therefore Ts stays in Q1 by Remark 6.5, so h(α2 − γ ) > 0. If α ≤ α ∗ , then M` is a sink or a weak sink, by Theorem 2.16; therefore Ts cannot converge to M` at −∞. By Theorem 2.19, there are no cycles in Q1 and no homoclinic orbits. By Remark 6.5, Ts cannot stay in Q1 ; hence Tr stays in Q1 and is bounded and converges at ∞ to M` . Thus h(α) < 0 for α1 < α ≤ α ∗ , so there exists at least an α crit ∈ (α ∗ , min(− p 0 , α2 ) such that h(α crit ) = 0. If it is not unique, we choose the smallest one. (ii) Let α > α ∗ . The existence and uniqueness of such a cycle in Q1 follows from Theorem 2.16 if α − α ∗ is small enough (Figure 15, lower left). For any α ∈ (α ∗ , α crit ), we still have existence: indeed, h(α) < 0 on this interval, so Tr stays in Q1 , and Tr cannot converge to M` at ∞, hence it has a limit cycle around M` at ∞. Since M` is a source, there also exist trajectories converging to M` at −∞, with a limit cycle at ∞. And Ts does not stay in Q1 , and it is bounded at −∞. Thus it has a limit cycle at −∞ surrounding (0, 0) and ±M` . (iii) Let α ≤ α ∗ (Figure 15, lower right). Then Tr stays in Q1 , is bounded on R, and converges to M` at ∞, while Ts does not stay in Q1 as above. Thus Ts has a limit cycle at −∞, containing the three stationary points. (iv) For any α < α crit apart from Tr and the cycles, all trajectories have a limit cycle at −∞ containing the three stationary points. By Theorem 2.21, all the cycles are contained in a ball B of R2 . Take any point P exterior to B. By Remark 6.5, T[P] has a limit cycle at −∞ contained in B and cannot have a limit cycle at ∞. Therefore y has constant sign near ln Sw . By Proposition 2.8, either Sw < ∞ or y is defined near ∞ and limτ →∞ ζ = L, lim r →∞ r α w = L. Finally we consider the case N /2 ≤ δ, where no cycle can exist.

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Theorem 6.9. Assume ε = −1 and α < 0 < N /2 ≤ δ < N . Then all solutions of (E w ) have a finite number of zeros, and w(r ) = `r −δ is a solution. If − p 0 ≤ α, Theorem 6.6 applies. If α < − p 0 , there exist positive solutions such that limr →0 r δ w = ` and lim r →∞ r η w = c > 0. All regular solutions have the same number m ≥ 2 of zeros. All other solutions satisfy lim r →−∞ r δ w = ±`, and have m or m + 1 zeros; there exist solutions with m + 1 zeros. Proof. By Proposition 2.11, all solutions have a finite number of zeros, and any solution is monotone near 0 and ln Sw , or converges to ±M` . By Remark 6.5, apart from Tr , all trajectories converge to ±M` at −∞. The functions V and W are nonincreasing. The trajectory Ts satisfies limτ →∞ V = limτ →∞ W = 0, so 0 V ≥ 0, W ≥ 0. If y has a zero at some point τ , then W (τ ) = −|Y (τ )| p / p 0 , which is impossible. If Y has a zero at some point θ , then V (θ) = −Y 0 (θ)2 /2, also a contradiction. Thus Ts stays in Q1 . By Remark 6.5 and Proposition 2.11, Tr does not stay in Q1 , but enters Q2 . By Lemma 6.7, Tr enters Q4 , and y has at least two zeros. Let m be the number of its zeros. Then Tr cuts the axis y = 0 at points (0, ξ1 ), . . . , (0, ξm ). Consider any trajectory T[P] with P = (0, ξ ), where ξ > |ξi | for 1 ≤ i ≤ m. It cannot intersect Tr or −Tr , so y has m + 1 zeros. Any trajectory has m or m + 1 zeros, because it does not meet Tr or −Tr or T[P] . And Sw < ∞ or lim r →∞ r α w = L 6= 0. Acknowledgment We thank Hector Giacomini (University of Tours) for interesting discussions on the existence of cycles in dynamical systems. References [Anderson and Leighton 1968] L. R. Anderson and W. Leighton, “Liapunov functions for autonomous systems of second order”, J. Math. Anal. Appl. 23 (1968), 645–664. MR 37 #5488 Zbl 0186.41602 [Barenblatt 1952] G. I. Barenblatt, “On self-similar motions of a compressible fluid in a porous medium”, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679–698. MR 14,699h Zbl 0047.19204 [Bidaut-Véron 1989] M.-F. Bidaut-Véron, “Local and global behavior of solutions of quasilinear equations of Emden–Fowler type”, Arch. Rational Mech. Anal. 107:4 (1989), 293–324. MR 90f: 35066 Zbl 0696.35022 [Bidaut-Véron 2006a] M. F. Bidaut-Véron, “The p-Laplace heat equation with a source term: selfsimilar solutions revisited”, Adv. Nonlinear Stud. 6:1 (2006), 69–108. MR MR2196892 [Bidaut-Véron 2006b] M. F. Bidaut-Véron, “Self-similar solutions of the p-Laplace heat equation: the case p > 2”, 2006. In preparation. [Chasseigne and Vazquez 2002] E. Chasseigne and J. L. Vazquez, “Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities”, Arch. Ration. Mech. Anal. 164:2 (2002), 133–187. MR 2003i:35148 Zbl 1018.35048


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[Chicone and Tian 1982] C. Chicone and J. H. Tian, “On general properties of quadratic systems”, Amer. Math. Monthly 89:3 (1982), 167–178. MR 84f:34044 Zbl 0466.34011 [DiBenedetto and Herrero 1990] E. DiBenedetto and M. A. Herrero, “Nonnegative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when 1 < p < 2”, Arch. Rational Mech. Anal. 111:3 (1990), 225–290. MR 92g:35088 [Gil and Vázquez 1997] O. Gil and J. L. Vázquez, “Focusing solutions for the p-Laplacian evolution equation”, Adv. Differential Equations 2:2 (1997), 183–202. MR 97g:35069 Zbl 1023.35514 [Hale and Koçak 1991] J. K. Hale and H. Koçak, Dynamics and bifurcations, Texts in Applied Mathematics 3, Springer, New York, 1991. MR 93e:58047 Zbl 0745.58002 [Hubbard and West 1995] J. H. Hubbard and B. H. West, Differential equations: a dynamical systems approach, v. 2: Higher dimensional systems, Texts in Applied Mathematics 18, Springer, New York, 1995. MR 96e:34001 Zbl 0824.34001 [Kamin and Vázquez 1992] S. Kamin and J. L. Vázquez, “Singular solutions of some nonlinear parabolic equations”, J. Anal. Math. 59 (1992), 51–74. MR 94e:35079 Zbl 0802.35066 [Qi and Wang 1999] Y.-W. Qi and M. Wang, “The global existence and finite time extinction of a quasilinear parabolic equation”, Adv. Differential Equations 4:5 (1999), 731–753. MR 2000d:35106 Zbl 0959.35104 [Vazquez and Véron 1996] J. L. Vazquez and L. Véron, “Different kinds of singular solutions of nonlinear parabolic equations”, pp. 240–249 in Nonlinear problems in applied mathematics: in honor of Ivar Stakgold on his 70th birthday, Soc. Ind. Appl. Math., Philadelphia, 1996. Zbl 0886.35078 [Zhao 1993] J. N. Zhao, “The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation”, J. Differential Equations 102:1 (1993), 33–52. MR 94c:35040 Zbl 0816.35070 Received February 4, 2005. Revised January 2, 2006. M ARIE F RANÇ OISE B IDAUT-V E´ RON L ABORATOIRE DE M ATHEMATIQUES ET P HYSIQUE T H E´ ORIQUE CNRS UMR 6083 FACULT E´ DES S CIENCES ET T ECHNIQUES PARC G RANDMONT 37200 T OURS F RANCE [email protected]


COMPLEXES OF TREES AND NESTED SET COMPLEXES E VA M ARIA F EICHTNER We exhibit an identity of abstract simplicial complexes between the wellstudied complex of trees Tn and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another generalization, more natural in the context of nested set complexes.

1. Introduction In this article we explore the connection between complexes of trees and nested set complexes of specific lattices. Nested set complexes appear as the combinatorial core in De Concini and Procesi’s [1995] wonderful compactifications of arrangement complements. They record the incidence structure of natural stratifications and are crucial for descriptions of topological invariants in combinatorial terms. Disregarding their geometric origin, nested set complexes can be defined for any finite meet-semilattice [Feichtner and Kozlov 2004]. Interesting connections between seemingly distant fields have been established when relating the purely order-theoretic concept of nested sets to various contexts in geometry. See [Feichtner and Yuzvinsky 2004] for a construction linking nested set complexes to toric geometry, and [Feichtner and Sturmfels 2005] for an appearance of nested set complexes in tropical geometry. This paper presents yet another setting where nested set complexes appear in a meaningful way and, this time, contribute to the toolbox of topological combinatorics and combinatorial representation theory. Complexes of trees Tn are abstract simplicial complexes with simplices corresponding to combinatorial types of rooted trees on n labeled leaves. They made their first appearance in [Boardman 1971], in connection with E ∞ -structures in homotopy theory. Later, they were studied in [Vogtmann 1990] from the point of MSC2000: primary 05E25; secondary 57Q05. Keywords: complexes of phylogenetic trees, nested set complexes, cohomology bases. 271

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view of geometric group theory, and in [Robinson and Whitehouse 1996] from the point of view of representation theory. In fact, Tn carries a natural action of the symmetric group 6n that allows for a lifting to a 6n+1 -action. For studying induced representations in homology, Robinson and Whitehouse determined the homotopy type of Tn to be a wedge of (n−1)! spheres of dimension n−3. Later on, complexes of trees were shown to be shellable by Trappmann and Ziegler [1998] and independently by Wachs (1998; see [Wachs 2003]). Recent interest in the complexes is motivated by the study of spaces of phylogenetic trees from combinatorial, geometric and statistics point of view [Billera et al. 2001]. Complexes of trees appear as links of the origin in natural polyhedral decompositions of the spaces of phylogenetic trees. Ardila and Klivans [2006] recently proved that the complex of trees Tn can be subdivided by the order complex of the partition lattice 1(5n ). Our result shows that 1(5n ), in fact, can be obtained by a sequence of stellar subdivisions from the complex of trees. This and other corollaries rely on the specific properties of nested set complexes that we introduce into the picture. Our paper is organized as follows: After recalling the definitions of complexes of trees and of nested set complexes in Section 2, we establish an isomorphism between the complex of trees Tn and the reduced minimal nested set complex of the partition lattice 5n in Section 3. Among several corollaries, we observe that the isomorphism provides a 6n -invariant approach for studying tree complexes; their 6n -representation theory can be retrieved literally for free. In Section 4 we complement the by now classical combinatorial correspondence between no broken circuit bases and decreasing EL-labeled chains for geometric lattices by incorporating proper maximal nested sets as defined in [De Concini and Procesi 2005]. We formulate a cohomology basis for the complex of trees that emerges naturally from this combinatorial setting. Mostly due to their rich representation theory, complexes of trees have been generalized to complexes of homeomorphically irreducible k-trees in [Hanlon 1996]. We discuss this and another, in the nested set context more natural, generalization in Section 5. 2. Main characters 2.1. The complex of trees. Let us fix some terminology: A tree is a cycle-free graph; vertices of degree 1 are called leaves of the tree. A rooted tree is a tree with one vertex of degree larger 1 marked as the root of the tree. Vertices other than the leaves and the root are called internal vertices. We assume that internal vertices have degree at least 3. The root of the tree is thus the only vertex that can have degree 2. Another way of saying this is that we assume all nonleaves to

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have outdegree at least 2, where the outdegree of a vertex is the number of adjacent edges that do not lie on the unique path between the vertex and the root. We call a rooted tree binary if the vertex degrees are minimal, i.e., the root has degree 2 and the internal vertices have degree 3; in other words, if the outdegree of all nonleaves is 2. Observe that a rooted binary tree on n leaves has exactly n−2 internal edges, i.e., edges that are not adjacent to a leave. The combinatorial type of a rooted tree with labeled leaves refers to its equivalence class under label- and root-preserving homeomorphisms of trees as 1-dimensional cell complexes. Rooted trees on n leaves labeled with integers 1, . . . , n are in one-to-one correspondence with trees on n+1 leaves labeled with integers 0, . . . , n where all internal vertices have degree ≥ 3. The correspondence is obtained by adding an edge and a leaf labeled 0 to the root of the tree. Though trees on n+1 labeled leaves seem to be the more natural, more symmetric objects, rooted trees on n leaves come in more handy for the description of Tn . Definition 2.1. The complex of trees Tn , n ≥ 3, is the abstract simplicial complex with maximal simplices given by the combinatorial types of binary rooted trees with n leaves labeled 1, . . . , n, and lower-dimensional simplices obtained by contracting at most n−3 internal edges. The complex of trees Tn is a pure (n−3)-dimensional simplicial complex. As we pointed out in the introduction, it is homotopy equivalent to a wedge of (n−1)! spheres of dimension n−3; see [Robinson and Whitehouse 1996, Theorem 1.5]. The complex T3 consists of 3 points. For n = 4, there are 4 types of trees, we depict labeled representatives in Figure 1. Observe that the first two correspond to 1-dimensional (maximal) simplices, whereas the other two correspond to vertices. The last two labeled trees, in fact, are the vertices of the edge corresponding to the first maximal tree. The third tree is a “vertex” of the second, and of the first. 2.2. Nested set complexes. We recall here the definition of building sets, nested sets, and nested set complexes for finite lattices as proposed in [Feichtner and Kozlov 2004].

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We use the standard notation for intervals in a finite lattice L, [X, Y ] := {Z ∈ L | X ≤ Z ≤ Y }, for X, Y ∈ L, moreover, S≤X := {Y ∈ S | Y ≤ X }, and accordingly SX , for S ⊆ L and X ∈ L. With max S we denote the set of maximal elements in S with respect to the order coming from L. Definition 2.2. Let L be a finite lattice. A subset G in L>0ˆ is called a building set if for any X ∈ L>0ˆ and max G≤X = {G 1 , . . . , G k } there is an isomorphism of partially ordered sets ϕX :

k Y

∼

= ˆ X] ˆ G j ] −→ [0, [0,

j=1

ˆ . . . , G j , . . . , 0) ˆ = G j for j = 1, . . . , k. We call FG (X ) := max G≤X with ϕ X (0, the set of factors of X in G. The full lattice L>0ˆ is the simplest example of a building set for L. We will sometimes abuse notation and just write L in this case. Besides this maximal building set, there is always a minimal building set I consisting of all elements X ˆ X ], in L>0ˆ which do not allow for a product decomposition of the lower interval [0, the so-called irreducible elements in L. Definition 2.3. Let L be a finite lattice and G a building set containing the maximal element 1ˆ of L. A subset S in G is called nested (or G-nested if specification is needed) if, for any set of incomparable elements X 1 , . . . , X t in S of cardinality at least two, the join X 1 ∨ · · · ∨ X t does not belong to G. The G-nested sets form an abstract simplicial complex, e N(L, G), the nested set complex of L with respect ˆ its base N(L, G) to G. Topologically, the nested set complex is a cone with apex 1; is called the reduced nested set complex of L with respect to G. We will mostly be concerned with reduced nested set complexes due to their more interesting topology. If the underlying lattice is clear from the context, we will write N(G) for N(L, G). Nested set complexes can be defined analogously for building sets not containˆ and, even more generally, for meet semilattices. For a definition in the full ing 1, generality, see [Feichtner and Kozlov 2004, Section 2]. For the maximal building set of a lattice L, subsets are nested if and only if they are linearly ordered in L. Hence, the reduced nested set complex N(L, L) coincides with the order complex of L, more precisely, with the order complex ˆ 1}, ˆ of L, which we denote by 1(L) using customary of the proper part, L \ {0, notation.


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If L is an atomic lattice, the nested set complexes can be realized as simplicial fans (see [Feichtner and Yuzvinsky 2004]), and for building sets G ⊆ H in L, the nested set complex e N(L, H) can be obtained from e N(L, G) by a sequence of stellar subdivisions [Feichtner and Müller 2005, Theorem 4.2]. In particular, any reduced nested set complex N(L, G) is obtained by a sequence of stellar subdivisions from the minimal reduced nested set complex N(L, I), and can be further subdivided by stellar subdivisions so as to obtain the maximal nested set complex 1(L). Example 2.4. Let 5n denote the lattice of set partitions of [n] := {1, . . . , n} partially ordered by reversed refinement. As explained above, the reduced maximal nested set complex N(5n , 5n ) is the order complex 1(5n ). Irreducible elements in 5n are the partitions with exactly one nonsingleton block. They can be identified with subsets of [n] of cardinality at least 2. Nested sets for the minimal building set I are collections of such subsets of [n] such that any two either contain one another or are disjoint. For n = 3, the reduced minimal nested set complex consists of 3 isolated points; for n = 4, it equals the Petersen graph. 3. Subdividing the complex of trees We now state the core fact of our note. Theorem 3.1. The complex of trees Tn and the reduced minimal nested set complex of the partition lattice N(5n , I) coincide as abstract simplicial complexes. Proof. We exhibit a bijection between simplices in Tn and nested sets in the reduced minimal nested set complex N(5n , I) of the partition lattice 5n . Let T be a tree in Tn with inner vertices t1 , . . . , tk . We denote the set of leaves in T below an inner vertex t by `(t). We associate a nested set S(T ) in N(5n , I) to T by defining S(T ) := {`(ti ) | i = 1, . . . , k}. Conversely, let S = {S1 , . . . , Sk } be a (reduced) nested set in 5n with respect to e(S) on the vertex set S ∪ {R}, where R will be the root I. We define a rooted tree T of the tree. Cover relations are defined by setting S > T if and only if T ∈ maxS Y ) := min(bX c \ bY c). This labeling in fact is an E L-labeling of L in the sense of [Björner and Wachs 1983], thus, by ordering maximal chains lexicographically, it induces a shelling of the order complex 1(L). Denote by dcω (L) the set of maximal chains in L with (strictly) decreasing label sequence: ˆ > λ(c2 > c1 ) > · · · > λ(1ˆ > cr −1 )}. dcω (L) := {0ˆ < c1 < · · · < cr −1 < 1ˆ | λ(c1 > 0) The characteristic cohomology classes [c∗ ] for c ∈ dcω (L), i.e., classes represented by cochains that evaluate to 1 on c and to 0 on any other top-dimensional simplex of 1(L), form a basis of the only nonzero reduced cohomology group of the order er −1 (1(L)). complex, H We add the notion of proper maximal nested sets to the standard notions for geometric lattices with fixed atom order that we listed so far. The concept has appeared in [De Concini and Procesi 2005]. Define a map φ : I → A(L) by setting φ(S) := minbSc for S ∈ I. A maximal nested set S in the (nonreduced) nested set complex e N(L, I) is called proper if the set {φ(S) | S ∈ S } is a basis of L. Denote the set of proper maximal nested sets in L by pnω (L). We define maps connecting nbcω (L), dcω (L), and pnω (L) for a given geometric lattice L. In the following proposition we will see that these maps provide bijective correspondences between the respective sets. To begin with, define 9 : nbcω (L) → dcω (L) by ˆ 9(a1 , . . . , ar ) = (0ˆ < ar < ar ∨ ar −1 < · · · < ar ∨ ar −1 ∨ · · · ∨ a1 = 1), where the a1 , . . . , ar are assumed to be in ascending order with respect to ω. Next, define 2 : dcω (L) → pnω (L) by (1)

ˆ = F(c1 ) ∪ F(c2 ) ∪ · · · ∪ F(cr −1 ) ∪ F(1), ˆ 2(0ˆ < c1 < · · · < cr −1 < 1)

for a chain c : 0ˆ < c1 < · · · < cr −1 < 1ˆ in L with decreasing label sequence, where F(ci ) denotes the set of factors of ci with respect to the minimal building set I in L. Finally, define 8 : pnω (L) → nbcω (L) by (2) for S ∈ pnω (L).

8(S) = {φ(S) | S ∈ S},


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Example 4.1. Consider the partition lattice 55 with lexicographic order lex on its set of atoms i j, 1 ≤ i < j ≤ 5. For b = {12, 14, 23, 45} ∈ nbclex (55 ) we have ˆ 9(b) = (0 < 45 < 23|45 < 23|145 < 1). Observe that the chain is constructed by taking consecutive joins of elements in the opposite of the lexicographic order. Going further using 2 we obtain the following proper nested set, ˆ = {45} ∪ {23, 45} ∪ {23, 145} ∪ {12345} 2(0 < 45 < 23|45 < 23|145 < 1) = {23, 45, 145, 12345}. Applying 8 we retrieve the no broken circuit basis we started with: 8({23, 45, 145, 12345}) = {12, 23, 14, 45}. Proposition 4.2. For a geometric lattice L with a given linear order ω on its atoms the maps 9, 2, and 8 defined above give bijective correspondences between (1) the no broken circuit bases nbcω (L) of L, (2) the maximal chains in L with decreasing label sequence, dcω (L), and (3) the proper maximal nested sets pnω (L) in e N(L, I), respectively. Proof. The map 9 : nbcω (L) → dcω (L) is well known in the theory of geometric lattices. It is the standard bijection relating no broken circuit bases to cohomology generators of the lattice; compare [Björner 1992, Section 7.6] for details. The composition of maps η := 2 ◦ 9 : nbcω (L) → pnω (L) is shown to be a bijection with inverse 8 : pnω (L) → nbcω (L); see [De Concini and Procesi 2005, Theorem 2.2]. This implies that 2 : dcω (L) → pnω (L) is bijective as well, which completes the proof of our claim. The aim of the next proposition is to trace the support simplices for the cohomology bases { [c∗ ] | c ∈ dcω (5n ) } of 1(5n ) through the inverse stellar subdivisions linking 1(5n ) to the complex of trees Tn = N(5n , I). For the moment we can stay with the full generality of geometric lattices and study support simplices for maximal simplices of 1(L) in the minimal reduced nested set complex N(L, I). Proposition 4.3. Let L be a geometric lattice, c : c1 < . . . < cr −1 a maximal simplex in 1(L). The maximal simplex in N(L, I) supporting c is given by the union of sets of factors F(c1 ) ∪ F(c2 ) ∪ · · · ∪ F(cr −1 ). Proof. There is a sequence of building sets L = G1 ⊇ G2 ⊇ · · · ⊇ Gt = I,

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connecting L and I which is obtained by removing elements of L \ I from L in nondecreasing order: Gi \ Gi+1 = {G i } with G i minimal in Gi \ I for i = 1, . . . , t−1. The corresponding nested set complexes are linked by inverse stellar subdivisions: N(Gi ) = st(N(Gi+1 ), V (FI (G i ))),

for i = 1, . . . t−1,

where V (FI (G i )) denotes the simplex in N(Gi+1 ) spanned by the factors of G i with respect to the minimal building set I. We trace what happens to the support simplex of c along the sequence of inverse stellar subdivisions connecting 1(L) with N(I). The support simplex of c remains unchanged in step i unless G i coincides with a (reducible) chain element c j (the irreducible chain elements can be replaced any time by their “factors”: F(ck ) = {ck } for ck ∈ I). We can assume that the support simplex of c in N(Gi ) is of the form S = F(c1 ) ∪ · · · ∪ F(c j−1 ) ∪ {c j } ∪ · · · ∪ {cr −1 },

and we aim to show that the support simplex of c in N(Gi+1 ) is given by T = F(c1 ) ∪ · · · ∪ F(c j−1 ) ∪ F(c j ) ∪ {c j+1 } ∪ · · · ∪ {cr −1 }.

Recall that the respective face posets of the nested set complexes are connected by a combinatorial blowup (3)

F(N(Gi )) = Bl F(G i ) (F(N(Gi+1 ))).

See [Feichtner and Kozlov 2004, 3.1] for the concept of a combinatorial blowup in meet semilattices. Hence, the support simplex S of c in N(Gi ) is of the form S = S0 ∪ {c j } with S0 ∈ N(Gi+1 ) (it is an element in the “copy” of the lower ideal of elements in F(N(Gi+1 )) having joins with F(G i )). Due to (3) we know that S0 6 ⊇ F(G i ) and S0 ∪ F(G i ) ∈ N(Gi+1 ), which in fact is the new support simplex of c. Let us mention in passing that, since we are talking about maximal simplices, S0 contains F(G i ) up to exactly one element X i ∈ L. Since c j is not contained in any of the F(ci ), i = 1, . . . , j−1, we have S0 = F(c1 ) ∪ · · · ∪ F(c j−1 ) ∪ {c j+1 } ∪ · · · ∪ {cr −1 },

and we find that T = F(c1 ) ∪ · · · ∪ F(c j−1 ) ∪ F(c j ) ∪ {c j+1 } ∪ · · · ∪ {cr −1 } as claimed. Example 4.4. Let us again consider the partition lattice 55 . The support simplex of c : 45 < 23|45 < 23|145 in N(55 , I) is {23, 45, 145}. We depict in Figure 3 how the support simplex of c changes in the sequence of inverse stellar subdivisions from 1(55 ) to N(55 , I).


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Figure 3. Support simplices of c. We now combine our findings to provide an explicit cohomology basis for the complex of trees Tn . We call a binary rooted tree T with n leaves labeled 1, . . . , n admissible, if, when recording the 2nd smallest label on the sets of leaves below any of the n−1 nonleaves of T , we find each of the labels 2, . . . , n exactly once. For an example of an admissible tree in T5 see Figure 4. Proposition 4.5. The characteristic cohomology classes associated with admissible trees in Tn , en−3 (Tn ) | T admissible in Tn }, { [T ∗ ] ∈ H form a basis for the (reduced) cohomology of the complex of trees Tn . en−3 (1(5n )) provided by characteristic Proof. We set out from the linear basis for H cohomology classes associated with the decreasing chains dcω (5n ) in 5n . Combining Proposition 4.3 with the definition of the bijection 2 : dcω (5n ) → pnω (5n ) in (1) we find that the characteristic cohomology classes associated with (reduced) en−3 (N(5n , I)). proper maximal nested sets pnω (5n ) provide a linear basis for H We tacitly make use of the bijection between maximal simplices in e N(L, I) and ˆ N(L, I) given by removing the maximal element 1 of L. 12345 23

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Figure 4. An admissible tree in T5 .

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To describe support simplices explicitly, recall that proper maximal nested sets are inverse images of no broken circuit bases under 8 : pnω (5n ) → nbcω (5n ) as defined in (2). The no broken circuit bases of 5n with respect to the lexicographic order on atoms, i.e., on pairs (i, j), 1 ≤ i < j ≤ n, are (n−1)-element subsets of the form (1, 2), (i 2 , 3), . . . , (i n−1 , n) with 1 ≤ i j ≤ j for j = 2, . . . , n−1. Inverse images under 8 are maximal nested sets S ∈ e N(5n , I) such that {φ(S) | S ∈ S} gives collections of pairs with each integer from 2 to n occurring exactly once in the second coordinate. Applying the isomorphism between N(5n , I) and Tn from Theorem 3.1 shows that the characteristic cohomology classes on admissible trees in Tn indeed form a basis of en−3 (Tn ). H Remark 4.6. Our basis of admissible trees differs from the one presented in [Trappmann and Ziegler 1998, Corollary 5] as a consequence of their shelling argument for complexes of trees. 5. Complexes of k-trees and other generalizations The intriguing representation theory of complexes of trees Tn in [Robinson and Whitehouse 1996] has given rise in [Hanlon 1996] to a generalization to complexes of k-trees. Definition 5.1. The complex of k-trees Tn(k) , n ≥ 1, k ≥ 1, is the abstract simplicial complex with faces corresponding to combinatorial types of rooted trees with (n−1)k+1 leaves labeled 1, . . . , (n−1)k+1, with all outdegrees at least k+1 and congruent to 1 modulo k, and at least one internal edge. The partial order among the rooted trees is given by contraction of internal edges. Alternatively, we could define Tn(k) as the simplicial complex with faces corresponding to (nonrooted) trees with (n−1)k+2 labeled leaves, all degrees of nonleaves at least k+2 and congruent to 2 modulo k, and at least one internal edge. Again, the order relation is given by contracting internal edges. Observe that for k = 1 we recover the complex of trees Tn . The face poset of our complex Tn(k) is the poset L(k) n−1 of [Hanlon 1996]. The complexes Tn(k) are pure simplicial complexes of dimension n−3. They are Cohen–Macaulay [Hanlon 1996, Theorem 2.3]; later, as mentioned in the introduction for the special case k = 1, a shellability result was obtained by Trappmann and Ziegler and, independently, by Wachs. The complexes Tn(k) carry a natural 6 N -action for N = (n−1)k+1 by permutation of leaves, which induces a 6 N -action on top degree homology. It follows from [Hanlon 1996, Theorem 1.1; Hanlon and Wachs 1995, Theorems 3.11 and


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en−3 (Tn(k) ) is isomorphic as an 6 N -module to H en−3 (5(k) ), where 5(k) 4.13] that H N N is the subposet of 5 N consisting of all partitions with block sizes congruent 1 modulo k. The poset 5(k) N had been studied before on its own right: it was shown to be Cohen–Macaulay in [Björner 1980], its homology and 6 N -representation theory was studied by Calderbank, Hanlon and Robinson [Calderbank et al. 1986]. en−3 (5(k) ) are isomorphic to the 1 N en−3 (Tn(k) ) and H In fact, both 6 N -modules H N homogeneous piece of the free Lie k-algebra constructed in [Hanlon and Wachs 1995]. This certainly provides enough evidence to look for a topological explanation of the isomorphism of 6 N -modules: Question 5.2. Is the complex of k-trees Tn(k) related to the order complex of (k) 5(k) (n−1)k+1 in the same way as Tn is related to the order complex of 5n , i.e., is Tn (k) homeomorphic to 1(5(k) (n−1)k+1 )? More than that, can 1(5(n−1)k+1 ) be obtained (k) from Tn by a sequence of stellar subdivisions? An approach to this question along the lines of Section 3 does not work right away: The poset 5(k) (n−1)k+1 is not a lattice, and a concept of nested sets for more general posets is not (yet) at hand. The generalization of Tn to complexes of k-trees Tn(k) thus turns out to be somewhat unnatural from the point of view of nested set constructions. We propose another generalization which is motivated by starting with a natural generalization of the partition lattice that remains within the class of lattices. Definition 5.3. For n > k ≥ 2, the k-equal lattice 5n,k is the sublattice of the partition lattice 5n that is join-generated by partitions with a single nontrivial block of size k. Observe that we retrieve 5n for k=2. There is an extensive study of the k-equal lattice in the literature, mostly motivated by the fact that 5n,k is the intersection lattice of a natural subspace arrangement, the k-equal arrangement. Its homology has been calculated in [Björner and Welker 1995], it was shown to be shellable in [Björner and Wachs 1996], and its 6n -representation theory has been studied in [Sundaram and Wachs 1997]. The irreducibles I in 5n,k are partitions with exactly one nontrivial block, this time of size at least k. Sets of irreducibles are nested if and only if for any two elements the nontrivial blocks are either contained in one another or disjoint. Constructing trees from nested sets, analogous to the construction of Tn from N(5n , I) in the proof of Theorem 3.1, suggests the following definition: Definition 5.4. The complex of k-equal trees Tn,k is a simplicial complex with maximal simplices given by combinatorial types of rooted trees T on n labeled

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leaves which are binary except at preleaves, where they are k-ary. Here, preleaves of T are leaves of the tree that is obtained from T by removing the leaves. Lowerdimensional simplices are obtained by contracting internal edges. We depict the tree types occurring as maximal simplices of T7,3 in Figure 5. Observe that one is a 3-dimensional simplex in T7,3 , whereas the two others are 2-dimensional. The definition of Tn,k does not appeal as natural, however, it is a trade off for the following Proposition and Corollary which are obtained literally for free, having the arguments of Section 3 at hand.

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Proposition 5.5. The complex of k-equal trees, Tn,k , and the minimal nested set complex of the k-equal lattice, N(5n,k , I), coincide as abstract simplicial complexes. In particular, the order complex 1(5n,k ) can be obtained from Tn,k by a sequence of stellar subdivisions. Proof. There is a bijection between trees in Tn,k and nested sets in N(5n,k , I) analogous to the bijection between Tn and N(5n , I) that we described in the proof of Theorem 3.1. Referring again to [Feichtner and Müller 2005, Theorem 4.2], the complexes are connected by a sequence of stellar subdivisions. Corollary 5.6. The graded homology groups of the complex of k-equal trees and of the k-equal lattice are isomorphic as 6n -modules: e∗ (Tn,k ) ∼ e∗ (5n,k ) H =6n H Note added in proof Question 5.2 has been answered affirmatively by Delucchi [2005]. Acknowledgments I thank Michelle Wachs and Federico Ardila for stimulating discussions at the IAS/Park City Mathematics Institute in July 2004. This work was carried out while the author was at the Swiss Federal Institute of Technology (ETH), Zurich, and at the Mathematical Sciences Research Institute, Berkeley.


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References [Ardila and Klivans 2006] F. Ardila and C. J. Klivans, “The Bergman complex of a matroid and phylogenetic trees”, J. Combin. Theory Ser. B 96:1 (2006), 38–49. MR 2006i:05034 Zbl 1082.05021 [Billera et al. 2001] L. J. Billera, S. P. Holmes, and K. Vogtmann, “Geometry of the space of phylogenetic trees”, Adv. in Appl. Math. 27:4 (2001), 733–767. MR 2002k:05229 Zbl 0995.92035 [Björner 1980] A. Björner, “Shellable and Cohen–Macaulay partially ordered sets”, Trans. Amer. Math. Soc. 260:1 (1980), 159–183. MR 81i:06001 Zbl 0441.06002 [Björner 1992] A. Björner, “The homology and shellability of matroids and geometric lattices”, pp. 226–283 in Matroid applications, edited by N. White, Encyclopedia Math. Appl. 40, Cambridge Univ. Press, Cambridge, 1992. MR 94a:52030 Zbl 0772.05027 [Björner and Wachs 1983] A. Björner and M. Wachs, “On lexicographically shellable posets”, Trans. Amer. Math. Soc. 277:1 (1983), 323–341. MR 84f:06004 Zbl 0514.05009 [Björner and Wachs 1996] A. Björner and M. Wachs, “Shellable nonpure complexes and posets, I”, Trans. Amer. Math. Soc. 348:4 (1996), 1299–1327. MR 96i:06008 Zbl 0857.05102 [Björner and Welker 1995] A. Björner and V. Welker, “The homology of “k-equal” manifolds and related partition lattices”, Adv. Math. 110:2 (1995), 277–313. MR 95m:52029 Zbl 0845.57020 [Boardman 1971] J. M. Boardman, “Homotopy structures and the language of trees”, pp. 37–58 in Algebraic topology (Madison, WI, 1970), edited by A. Liulevicius, Proc. Sympos. Pure Math. 22, Amer. Math. Soc., Providence, RI, 1971. MR 50 #3215 Zbl 0242.55012 [Calderbank et al. 1986] A. R. Calderbank, P. Hanlon, and R. W. Robinson, “Partitions into even and odd block size and some unusual characters of the symmetric groups”, Proc. London Math. Soc. (3) 53:2 (1986), 288–320. MR 87m:20042 Zbl 0602.20017 [De Concini and Procesi 1995] C. De Concini and C. Procesi, “Wonderful models of subspace arrangements”, Selecta Math. (N.S.) 1:3 (1995), 459–494. MR 97k:14013 Zbl 0842.14038 [De Concini and Procesi 2005] C. De Concini and C. Procesi, “Nested sets and Jeffrey-Kirwan residues”, pp. 139–149 in Geometric methods in algebra and number theory, edited by F. Bogomolov and Y. Tschinkel, Progress in Math. 235, Birkhäuser, Boston, 2005. MR 2006j:32032 Zbl 1093.52503 [Delucchi 2005] E. Delucchi, “Subdivision of complexes of k-trees”, preprint, ETH Zurich, 2005. math.CO/0509378 [Feichtner and Kozlov 2004] E.-M. Feichtner and D. N. Kozlov, “Incidence combinatorics of resolutions”, Selecta Math. (N.S.) 10:1 (2004), 37–60. MR MR2061222 Zbl 1068.06004 [Feichtner and Müller 2005] E. M. Feichtner and I. Müller, “On the topology of nested set complexes”, Proc. Amer. Math. Soc. 133:4 (2005), 999–1006. MR 2006c:06005 Zbl 1053.05125 [Feichtner and Sturmfels 2005] E. M. Feichtner and B. Sturmfels, “Matroid polytopes, nested sets and Bergman fans”, Port. Math. (N.S.) 62:4 (2005), 437–468. MR 2006j:05036 Zbl 1092.52006 [Feichtner and Yuzvinsky 2004] E. M. Feichtner and S. Yuzvinsky, “Chow rings of toric varieties defined by atomic lattices”, Invent. Math. 155:3 (2004), 515–536. MR 2004k:14009 Zbl 1083.14059 [Hanlon 1996] P. Hanlon, “Otter’s method and the homology of homeomorphically irreducible ktrees”, J. Combin. Theory Ser. A 74:2 (1996), 301–320. MR 98g:05155 Zbl 0848.05021 [Hanlon and Wachs 1995] P. Hanlon and M. Wachs, “On Lie k-algebras”, Adv. Math. 113:2 (1995), 206–236. MR 96h:17006 Zbl 0844.17001 [Oxley 1992] J. G. Oxley, Matroid theory, Oxford University Press, New York, 1992. MR 94d:05033 Zbl 0784.05002

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[Robinson and Whitehouse 1996] A. Robinson and S. Whitehouse, “The tree representation of 6n+1 ”, J. Pure Appl. Algebra 111:1-3 (1996), 245–253. MR 97g:55010 Zbl 0865.55010 [Stanley 1982] R. P. Stanley, “Some aspects of groups acting on finite posets”, J. Combin. Theory Ser. A 32:2 (1982), 132–161. MR 83d:06002 Zbl 0496.06001 [Sundaram and Wachs 1997] S. Sundaram and M. Wachs, “The homology representations of the kequal partition lattice”, Trans. Amer. Math. Soc. 349:3 (1997), 935–954. MR 97j:05063 Zbl 0863. 05082 [Trappmann and Ziegler 1998] H. Trappmann and G. M. Ziegler, “Shellability of complexes of trees”, J. Combin. Theory Ser. A 82:2 (1998), 168–178. MR 99f:05122 Zbl 0916.06004 [Vogtmann 1990] K. Vogtmann, “Local structure of some Out(Fn )-complexes”, Proc. Edinburgh Math. Soc. (2) 33:3 (1990), 367–379. MR 92d:57002 Zbl 0694.20021 [Wachs 2003] M. Wachs, “Posets of graphs, partitions and trees”, talk at Banff International Research Station, May 2003, Available at http://www.math.miami.edu/~wachs/talks.html. Based on joint work with John Shareshian. Received February 5, 2005. E VA M ARIA F EICHTNER D EPARTMENT OF M ATHEMATICS U NIVERSITY OF S TUTTGART 70569 S TUTTGART G ERMANY [email protected]


INVERSION INVARIANT ADDITIVE SUBGROUPS OF DIVISION RINGS DANIEL G OLDSTEIN , ROBERT M. G URALNICK , L ANCE S MALL AND E FIM Z ELMANOV We characterize the inversion invariant additive subgroups of any field, and, more generally, those of a division ring (apart from division rings of characteristic 2). We also show how a classical identity of Hua provides a bridge between this problem and Jordan algebras.

Answering a question of Dan Mauldin, we characterize in this note the inversion invariant additive subgroups of a field. We show in Section 2 that aside from the case of imperfect fields of characteristic 2, a nonzero subgroup that is inversion invariant is either a subfield or the set of trace-zero elements in a subfield with respect to an automorphism of order 2. A key ingredient in the proof is a simple, classical identity of Hua, through which we can bring to bear known results about Jordan algebras and Jordan triple systems. We also solve in Section 1 the same problem for division rings of characteristic not 2. 1. Division rings We characterize pairs T ⊆ D, where D is a division ring of characteristic 6= 2; T is an additive of subgroup of D; for an arbitrary nonzero element of t ∈ T the inverse t −1 again lies in T , and T generates D as a ring. Example 1. T = D. Example 2. Let σ : D → D be an involution of a division ring D (an additive map such that σ (σ (a)) = a and σ (ab) = σ (b)σ (a) for arbitrary elements a, b in D). Then T = H (D, σ ) = {a ∈ D | σ (a) = a} is inversion invariant. The additive subgroup H (D, σ ) generates D unless D is a quaternion algebra over its center and σ is a quaternionic involution; see [Herstein 1976]. MSC2000: 17C10, 17A35, 12E99. Keywords: Jordan algebra, Clifford algebra, Jordan triple system. Guralnick was supported by the National Science Foundation grant DMS 0140578, and Zelmanov was supported by NSF grant DMS 0350399. 287

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Example 3. The additive group of skew symmetric elements S(D, σ ) = {a ∈ D | σ (a) = −a} is also inversion invariant. It generates D unless D is a field and σ = Id. Example 4. Let σ, τ be two commuting involutions of a division ring D, and set T = H (D, σ ) ∩ S(D, τ ). The subgroup T is inversion invariant. Example 5. Let g be an automorphism of a division ring D of order 2. Then T = S(D, g) = {a ∈ D | g(a) = −a} is inversion invariant and generates D. Finally, we will describe somewhat special pairs which we will refer to as Clifford pairs. Let V be a vector space over a field F with a quadratic form q : V → F such that the Clifford algebra C = C(V, q) is a division ring. Let J = F · 1 + V ⊆ C and let 0 6= x ∈ C be an element such that x J x = J . Then (i)

T = xJ ⊂C

is inversion invariant and generates C. Now suppose that the space V is finite dimensional over F and dim F V = n is even. Choose an orthogonal basis v1 , . . . , vn of V . Let 0 6= x ∈ C be an element such that x(J + Fv1 · · · vn )x = J + Fv1 · · · vn . Then (ii)

T = x(J + Fv1 · · · vn ) ⊂ C

is inversion invariant. Let hT ieven , hT iodd be the additive subgroups of C generated by products of elements of T of even and odd lengths, respectively. In the pairs of types (i) and (ii) hT ieven = hT iodd . This does not always have to be the case. Let D be a Z/2Z-graded division ring, D = D0 + D1 , whose even part D0 is isomorphic to the Clifford algebra C of a quadratic form q : V → F. Suppose x ∈ D1 satisfies x J x = J . Then (iii)

T = x J ⊂ D1 ⊂ D

is inversion invariant and generates D. If dim F V = n is even, let v1 , . . . , vn be an orthogonal basis of V . If 0 6= x ∈ D1 satisfies x(J + Fv1 · · · vn )x = J + Fv1 · · · vn , then (iv)

T = x(J + Fv1 · · · vn ) ⊂ D1 ⊂ D

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is inversion invariant and generates D. In the pairs of types (iii) and (iv) hT ieven ∩ hT iodd = 0. Theorem 1.1. If T is an inversion invariant additive subgroup of a division ring D, the characteristic of D is not 2, and T generates D, then T ⊆ D is a pair of one of the types described in Examples 1–5 or of one of the Clifford types (i)–(iv). The key observation is an elementary identity of L. K. Hua [Jacobson 1980, Exercise 9, page 92]. Lemma 1.2 (Hua’s Identity). Let R be a ring with elements a, b ∈ R such that a, b and ab − 1 are invertible. Then −1 = aba − a. (a − b−1 )−1 − a −1 Corollary 1.3. If T is an inversion invariant additive subgroup of a division ring D then for arbitrary elements a, b ∈ T we have aba ∈ T . We will need a few definitions from the theory of Jordan systems. A (not necessarily associative) ring J is called a Jordan ring if it satisfies (J1)

x y = yx,

(J2)

(x y)x = x 2 (yx), 2

for all x and y in J . Example 6. If R is an associative ring then the additive group of R with the new multiplication x · y = x y + yx is a Jordan ring which is denoted R (+) . Example 7. If σ : R → R is an involution, then H (R, σ ) = {a ∈ R | σ (a) = a} is a Jordan subring of R (+) . Example 8. Let V be a vector space over a field F, q : V → F a quadratic form, and let C = C(V, q) be the Clifford algebra. The space J = F · 1 + V is a Jordan subalgebra of C (+) , which is called the Jordan algebra of the quadratic form q. A Jordan ring J is said to be special if it is embeddable into R (+) , where R is an associative ring. If R is generated by J then R is an associative enveloping ring of R. Clearly, Examples 6, 7, and 8 are special. A Jordan ring that is not special is called exceptional. Theorem 1.4. A simple Jordan ring of characteristic 6= 2 is either isomorphic to R (+) , where R is a simple associative ring, or to H (R, σ ), where R is a simple associative ring with an involution σ , or to a Jordan algebra of a nondegenerate quadratic form or is an exceptional 27-dimensional algebra over its center (called an Albert algebra).

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Simple (quadratic) Jordan algebras of characteristic 2 were classified in [McCrimmon and Zelmanov 1988]. Let T be an additive group equipped with group endomorphisms P(x) for each x in T . We assume that the map from x to P(x) is quadratic; that is, P(nx) = n 2 P(x) and the map P(x1 + x2 ) − P(x1 ) − P(x2 ) is bilinear in x1 and x2 . In other words, the product P(x)y is quadratic in x and linear in y. We write {x, y, z} = P(x + z)y − P(x)y − P(z)y. We say that T is a Jordan triple system if it satisfies the identities (JT1) (JT2) (JT3)

{z, P(x)y, y} = {z, x, P(y)x}, P(P(x)y)z = P(x)P(y)P(x)z, {P(x)z, y, x} = P(x){z, x, y}

and all their linearizations (i.e., identities of lower degree coming from bilinearity: these are superfluous if we are working over a field F of at least four elements). Example 9. If R is an associative ring then the additive group of R with a product P(x)y = x yx is a Jordan triple system which we will also denote by R (+) . Any Jordan triple system T gives rise to a family of Jordan rings T (a) , the ahomotopes, by fixing the middle element: x · y = {x, a, y}. A Jordan triple system T is called special if it is embeddable into R (+) , where R is an associative ring, T ⊆ R (+) . If the ring R is generated by T then we say R is an associative enveloping algebra of T . An associative enveloping algebra R of T is said to be tight if T ∩ I 6= (0) for any nonzero ideal I of R. Let G be the ideal of the free Jordan triple system constructed in [Zelmanov 1984]. The identities from G distinguish triples of Clifford type assuming that the characteristic 6= 2, 3. Similar ideals for characteristics 2 and 3 were constructed in [D’Amour and McCrimmon 2000]. Proposition 1.5 [Zelmanov 1984]. Let T be a simple special Jordan triple system such that G(T ) 6= (0). Then T is isomorphic to one of the triples R (+) ; H (R, σ ) = {a ∈ R | σ (a) = a}; S(R, σ ) = {a ∈ R | σ (a) = −a}; H (R, σ )∩ S(R, τ ); S(R, g), where R is a simple associative ring; σ, τ are involutions, σ τ = τ σ ; g is an automorphism of order 2. Moreover, the ring R is in some sense unique. Proposition 1.6 [Zelmanov 1984]. Let T1 , T2 be simple Jordan triple systems, such that G(Ti ) 6= (0). Let Ti ⊆ Ri be their tight associative enveloping algebras. Then any isomorphism T1 → T2 can be extended to an isomorphism or anti-isomorphism R1 → R2 .


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Proposition 1.7 [Zelmanov 1984]. Let T be a simple special Jordan triple system such that G(T ) = (0). Then T is isomorphic to a mutation or a polarization of a Jordan triple system of a quadratic form. In particular, an arbitrary homotope T (a) is the Jordan algebra of a quadratic form. Proof of Theorem 1.1. Let T ⊆ D be an inversion invariant additive subgroup that generates D. By the corollary to Hua’s identity, T is a Jordan subtriple of D (+) and D is a tight associative enveloping algebra of T . If G(T ) 6= (0), Propositions 1.5 and 1.6 imply that T ⊆ D is one of the types in Examples 1–5. Suppose now that G(T ) = (0). Choose an arbitrary nonzero element x ∈ T . Then J = x −1 T is a unital Jordan subalgebra of D (+) . Indeed if a, b, ∈ T then (x −1 a)(x −1 b)+(x −1 b)(x −1 a) = x −1 (ax −1 b+bx −1 a) ∈ x −1 T = J . By Proposition 1.7, J is isomorphic to the Jordan algebra of a symmetric nondegenerate bilinear form, J = F · 1 + V , where F is a subfield of D, V is an F-space; vw + wv = q(v, w)1; v, w, ∈ V and q(v, w) ∈ F. Case 1. J generates D and the dimension dim F V is either infinite or even. In this case the only associative enveloping algebra of J is the Clifford algebra C(V, q) (see [Jacobson 1968]). Hence D ∼ = C(V, q), T = x J and T ⊆ D is a pair of type (i). Case 2. J generates D and the dimension dim F V is odd. Choose an orthogonal basis v1 , . . . , vn+1 of V . Then the element z = v1 · · · vn+1 lies in the center of the Clifford algebra C(V, q), where z 2 = ±q(v1 ) · · · q(vn+1 ) depending on whether or not n is divisible by 4. If z 2 is not the square of an element of F then C(V, q) is simple. In this case, D ∼ = C(V, q) and T ⊆ D is again a pair of type (i). 2 Suppose now that z = α 2 , α ∈ F. Then z = α or −α in D and vn+1 ∈ Fv1 · · · vn . Pn Let V 0 = i=1 Fvi . We have D ∼ = C(V 0 , q), J = F · 1 + V 0 + Fv1 · · · vn , and T = x J ⊆ D is a pair of type (ii). Case 3. The subring hJ i of D generated by J is not equal to D and the dimension dim F V is infinite or even. Then D = D0 + D1 , where D0 = hJ i, D1 = xhJ i is a Z/2Z-grading. As above, D0 ∼ = C(V, q) and the pair T ⊆ D is of type (iii). Case 4. hJ i 6= D and dim F V is odd. Choose an orthogonal basis v1 , . . . , vn+1 of Pn V . Set V 0 = i=1 Fvi and z = v1 · · · vn+1 . If z 2 is not equal to the square of an element of F then hJ i ∼ = C(V, q) and T ⊂ D is of type (i). Otherwise, vn+1 ∈ Fv1 · · · vn lies in D and T = x(F · 1 + V 0 + Fv1 · · · vn ) ⊆ D1 ⊆ D is a pair of type (iv). This finishes the proof of the proposition. 2. Fields In this section, we classify the inversion invariant additive subgroups of fields. Of course this result is included in the main theorem from the previous section as long as the characteristic is not 2. We include the proof for all characteristics.

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Let K be a field. Write k for the prime subfield of K . Let V be an inversion invariant additive subgroup of K . Since V ⊆ k(V ), there is no harm in assuming that K = k(V ). If the characteristic p of the field L is positive, write L p = {x p | x ∈ L} for the subfield consisting of p-th powers of elements of L. Our main result for the commutative case is: Theorem 2.1 (Commutative Theorem). Let K be a field of characteristic p ≥ 0. Let V be a nonzero subset of K that is a subgroup under addition. Let k be the prime field of K and assume that K = k(V ). Then V is closed under inversion if and only if one of the following holds: (1) p 6= 2 and either V = K or there is an automorphism σ of K of order 2 such that V = {x ∈ K | σ (x) = −x}; (2) p = 2 and V is a K 2 -module. It is straightforward to verify that the possibilities mentioned in the theorem are inversion invariant. In particular, if K has characteristic 2, then (a K 2 )−1 = a −1 K 2 = a(a −2 K 2 ) = a K 2 and so any K 2 -submodule of K is inversion invariant. Corollary 2.2. Let K be a field of characteristic p ≥ 0. Assume that 1 ∈ V and that either p 6= 2 or that the field generated by V is perfect (in particular, if K is contained in the algebraic closure of the field of size 2). Then V is inversion invariant if and only if V is a subfield of K . The remainder of this section is devoted to the proof of Theorem 2.1. Lemma 2.3. Let K be a field and V an additive subgroup of K closed under inversion. Let 0 6= a ∈ V . Then a 3 ∈ V and a 2 V = V . Proof. We saw in the corollary to Hua’s identity that if a, b in V , then aba ∈ V . Taking b = a, we conclude that a 3 ∈ V . We also have a 2 V ⊆ V . We get equality by multiplying by a −1 . Let K be a field and V a nonzero additive subgroup of K closed under inversion. Let V2 be the set of all products x y with x, y ∈ V . We have the following lemma. Lemma 2.4. Assume that p 6= 2. Let 0 6= a ∈ V . Then (1) V2 = V a;

(2) V2 a = V ;

(3) V a 2 = V.

Proof. We have already seen that a 2 V ⊆ V . Thus, 2ab = (a + b)2 − a 2 − b2 also sends V back to itself for any a, b ∈ V . Since the characteristic of K is not 2, the identity v/2 = 1/(1/v + 1/v) for nonzero v shows that (1/2)V ⊆ V . Hence abV ⊆ 2abV ⊆ V . If a 6= 0, then using a −1 instead of a gives V b ⊆ V a.


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Hence V2 ⊆ V a since b was arbitrary in V . The reverse inclusion follows from the definition of V2 . This proves (1). Taking a −1 for a gives (2), and (3) follows from (1) and (2). We now complete the proof in characteristic not 2. Theorem 2.5. Let K be a field of characteristic not 2 with V a nontrivial additive subgroup of K closed under inversion. Then (1) V2 is a field. (2) The ring R generated by V (over Z) is a field. (3) Either R = V = V2 or [R : V2 ] = 2. Proof. We have V2 = V a for any nonzero a ∈ V . Fix such an a. Thus, V2 V2 = V aV a = V V a 2 = V V = V2 . Hence V2 is a subgroup under addition and closed under multiplication. Since it also closed under inverses, it follows that V2 is a field. This proves (1). Let R = Z[V ] be the ring generated by V . Since V2 V = V and V V = V2 , we see that R = V + V2 . Thus, R is a field (since it is algebraic over the field V2 and is contained in the field K ). This proves (2). We have already seen that R = V + V2 and that V and V2 are each onedimensional modules over the field V2 . So either the sum is direct and [R : V2 ] = 2, or R = V = V2 . This gives the main result for fields of characteristic not 2 — for either V itself is a field or the ring R generated by V is a field and [R : V2 ] = 2. We see that V = aV2 with a 2 ∈ V2 , whence V is the −1 eigenspace of a nontrivial automorphism of R/V2 . It remains to prove Theorem 2.1 in the case that the field K has characteristic 2. We have a 2 b ∈ V whenever a, b in V . It follows that V is a k[V 2 ] module. Set J = k[V 2 ]. So J a −1 ⊆ V if 0 6= a ∈ V . Therefore J −1 a ⊆ V since V is inversion invariant. This shows that V is in fact a module over k(V 2 ) = K 2 and completes the proof of Theorem 2.1. Also note that since V is a K 2 -module, the same is true for the ring R generated by V , whence this ring is a field. Remark 2.6. From the arguments above, it follows that if T is a nonzero additive subgroup of a division ring D that is closed under inversion and either the characteristic is not 2 or D is a field, then the ring generated by T (over Z) is a division ring. References [D’Amour and McCrimmon 2000] A. D’Amour and K. McCrimmon, “The structure of quadratic Jordan systems of Clifford type”, J. Algebra 234 (2000), 31–89. MR 2001k:17050 Zbl 0982.17015

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[Herstein 1976] I. N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976. MR 56 #406 Zbl 0343.16011 [Jacobson 1968] N. Jacobson, Structure and representations of Jordan algebras, AMS Colloquium Publications 39, Amer. Math. Soc., Providence, RI, 1968. MR 40 #4330 Zbl 0218.17010 [Jacobson 1980] N. Jacobson, Basic algebra, vol. 2, Freeman, San Francisco, 1980. MR 81g:00001 Zbl 0441.16001 [McCrimmon and Zelmanov 1988] K. McCrimmon and E. Zelmanov, “The structure of strongly prime quadratic Jordan algebras”, Adv. in Math. 69:2 (1988), 133–222. MR 89k:17052 Zbl 0656.17015 [Zelmanov 1984] E. I. Zelmanov, “Prime Jordan triple systems, II”, Sibirsk. Mat. Zh. 25:5 (1984), 50–61. In Russian; translated in Siberian Math. J. 25:5 (1984), 726–735. MR 87d:17022 Zbl 0562.17007 Received January 13, 2005. DANIEL G OLDSTEIN C ENTER FOR C OMMUNICATIONS R ESEARCH S AN D IEGO , CA 92121-1969 U NITED S TATES [email protected] ROBERT M. G URALNICK D EPARTMENT OF M ATHEMATICS U NIVERSITY OF S OUTHERN C ALIFORNIA 3620 S. V ERMONT AVE . L OS A NGELES , CA 90089-2532 U NITED S TATES [email protected] L ANCE S MALL D EPARTMENT OF M ATHEMATICS U NIVERSITY OF C ALIFORNIA S AN D IEGO L A J OLLA , CA 92093-0112 U NITED S TATES [email protected] E FIM Z ELMANOV D EPARTMENT OF M ATHEMATICS U NIVERSITY OF C ALIFORNIA S AN D IEGO L A J OLLA , CA 92093-0112 U NITED S TATES [email protected]


MANIFOLDS WITH POSITIVE SECOND GAUSS–BONNET CURVATURE M OHAMMED -L ARBI L ABBI The second Gauss–Bonnet curvature of a Riemannian manifold, denoted h4 , is a generalization of the four-dimensional Gauss–Bonnet integrand to higher dimensions. It coincides with the second curvature invariant, which appears in the well known Weyl’s tube formula. A crucial property of h4 is that it is nonnegative for Einstein manifolds; hence it provides, independently of the sign of the Einstein constant, a geometric obstruction to the existence of Einstein metrics in dimensions ≥ 4. This motivates our study of the positivity of this invariant. We show that positive sectional curvature implies the positivity of h4 , and so does positive isotropic curvature in dimensions ≥ 8. Also, we prove many constructions of metrics with positive second Gauss–Bonnet curvature that generalize similar well known results for the scalar curvature.

1. Introduction and statement of the results Let (M, g) be a smooth Riemannian manifold of dimension n ≥ 4. Let R, c R and c2 R denote respectively the Riemann curvature tensor, Ricci tensor and the scalar curvature of (M, g). The second Gauss–Bonnet curvature, which throughout this paper is abbreviated as SGBC and denoted by h 4 , is a generalization of the four-dimensional Gauss–Bonnet integrand to higher dimensions. It is a scalar expression that is quadratic in the curvature tensor. It can be defined by h 4 = kRk2 − kc Rk2 + 41 kc2 Rk2 . A crucial property of h 4 is that it is nonnegative for Einstein manifolds (see Section 3 below), and so it provides a new geometric obstruction to the existence of Einstein metrics independently of the sign of the Einstein constant. In particular, the manifolds that do not admit any metric with positive SGBC cannot admit any Einstein metric. MSC2000: 53C21, 53B20. Keywords: Gauss–Bonnet curvature, Einstein manifold, surgery. The author thanks the referee for useful comments and for remarks about terminology on the first version of this paper. 295

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Recall that in dimensions greater than 4, we do not know any topological restriction for a manifold to be Einstein. If one requires the Einstein constant to be positive, then one has two geometric obstructions c R > 0 and c2 R > 0. It would then be a great benefit to have a classification of manifolds with positive SGBC. In this paper, we inaugurate the study of the positivity properties of this important invariant. The paper is divided into five sections. In Section 2, we use the ring of curvature structures to introduce and study all of the Gauss–Bonnet curvatures (h 2k ). They are, up to a constant, the curvature invariants that appear in Weyl’s tube formula. Many examples are included. In Section 3, we study separately the case of the second invariant. We prove that it is nonnegative for Einstein manifolds, and nonpositive for conformally flat manifolds with zero scalar curvature. The limit cases are discussed. Also, we prove the following theorem: Theorem A. Let (M, g) be a Riemannian manifold of dimension n ≥ 4, with nonnegative p-curvature, such that p ≥ n/2. Then the SGBC of (M, g) is nonnegative. Furthermore, it vanishes if and only if the manifold is flat. The same statements hold with “nonnegative” replaced by “positive”. In particular, positive sectional curvature implies positive SGBC. Also, if n ≥ 8, positive isotropic curvature implies positive SGBC. The same statements hold with “positive” replaced by “nonnegative”. Therefore we can apply our previous constructions in the class of manifolds with positive p-curvature (see Labbi [1997a; 1997b; 2000]) to get many examples of metrics with positive SGBC. In Section 4, we prove the following useful theorem, which generalizes a similar result for the scalar curvature. Theorem B. Suppose that the total space M of a Riemannian submersion is compact, and the fibers (with the induced metric) have positive SGBC. Then the manifold M admits a Riemannian metric with positive SGBC. We end the section with two applications of this theorem. In Section 5, we prove the following stability theorem in the class of compact manifolds with positive SGBC: Theorem C. If a manifold M is obtained from a compact manifold X by surgery in codimension ≥ 5, and X admits a metric of positive SGBC, then so does M. In particular, the connected sum of two compact manifolds of dimensions ≥ 5, each one having positive SGBC, admits a metric with positive SGBC.

POSITIVE SECOND GAUSS–BONNET CURVATURE

297

This generalizes a celebrated theorem of Gromov–Lawson and Schoen–Yau for the scalar curvature. As a consequence of Theorem C, we prove that there are no restrictions on the fundamental group of a compact manifold of dimension ≥ 6 to carry a metric with positive SGBC. Finally, we mention that it would be interesting to prove, as in the case of the scalar curvature, that every manifold with nonnegative SGBC not identically zero admits a metric with positive SGBC. 2. The Gauss–Bonnet curvatures The Gauss–Bonnet curvatures are conveniently manipulated with the use of the ring of curvature structures. Let us first recall some properties of this ring. V V L Let ∗ M = p≥0 ∗ p M denote the ring of differential forms on M, where M is as above. Considering the tensor product over the ring of smooth functions, we V V V V L define D = ∗ M ⊗ ∗ M = p,q≥0 D p,q , where D p,q = ∗ p M ⊗ ∗q M. It is a graded associative ring and called the ring of double forms on M. The Kulkarni– Nomizu product in D shall be denoted by a dot, omitted whenever possible. P The ring of curvature structures on M [Kulkarni 1972] is the ring C = p≥0 C p , where C p denotes symmetric elements in D p, p . We denote by C1 , C2 , C0 the subring of curvature structures satisfying, respectively, the first, the second, and both the first and second Bianchi identity. V The standard inner product and the Hodge star operator ∗ on ∗ p M can be extended in a standard way to D. These extensions were used in [Labbi 2005] to prove many properties of the former ring. In particular, it is proved that gω = ∗ c ∗ ω, for all ω ∈ D, where c denotes the contraction map. Also, for all ω1 , ω2 ∈ D, we have h gω1 , ω2 i = hω1 , cω2 i; that is, the contraction map is the formal adjoint of the multiplication map by the metric g. We have (1)

hω1 , ω2 i = ∗(ω1 .∗ ω2 ) = ∗ (∗ ω1 .ω2 ),

for all ω1 , ω2 ∈ D p,q , and ∗∗ = (−1)( p+q)(n− p−q) Id, where Id is the identity map on D p,q . Next, we define the Gauss–Bonnet curvatures:

298


Definition. The 2q-Gauss–Bonnet curvature, denoted h 2q , is the complete contraction of the tensor R q , precisely, h 2q =

1 c2q R q , (2q)!

where R q denotes the multiplication of R with itself q times in the ring C. Note that h 2 = 21 c2 R is one half of the scalar curvature, and if n is even, then h n is (up to a constant) the Gauss–Bonnet integrand. Finally, note that in [Labbi 2005], it is proved that (2)

h 2q = ∗

1 g n−2q R q . (n − 2q)!

Example 2.1. Let (M, g) have constant sectional curvature λ. Then R = 21 λg 2

Rq =

and

λq 2q g . 2q

Therefore, h 2q is constant: h 2q = ∗

λq λq n! 1 g n−2q R q = ∗ q gn = q . (n − 2q)! 2 (n − 2q)! 2 (n − 2q)!

In particular, n(n − 1)(n − 2)(n − 3) 2 λ . 4 Example 2.2. Let (M, g) be a Riemannian product of two Riemannian manifolds (M1 , g1 ) and (M2 , g2 ). If we index by i the invariants of the metric gi , for i = 1, 2, then q X q−i q R = R1 + R2 and R q = (R1 + R2 )q = Ci R1i R2 . h4 =

i=0

A straightforward calculation then shows that q

h 2q =

=

c2q R q X q c2q q−i = Ci (R i R ) (2q)! (2q)! 1 2 q X i=0

i=0 i 2q−2i q−i 2i R2 q c R1 c Ci

(2i)! (2q − 2i)!

=

q X

q

Ci (h 2i )1 (h 2q−2i )2 .

i=0

In particular, (3)

h 4 = (h 4 )1 + 12 scal1 scal2 + (h 4 )2 ,

where scal denotes the scalar curvature.

POSITIVE SECOND GAUSS–BONNET CURVATURE

299

Example 2.3. Let (M, g) be a hypersurface of the Euclidean space. If B denotes the second fundamental form at a given point, then the Gauss equation shows that R = 21 B 2

Rq =

and

1 2q B . 2q

Consequently, if λ1 ≤ λ2 ≤ · · · ≤ λn denote the eigenvalues of B, then the eigenvalues of R q are ((2q)!/2q ) λi1 , λi2 , . . . , λi2q , where i 1 < · · · < i 2q . Hence, h 2q =

(2q)! 2q

X

λi1 . . . λi2q .

1≤i 1 0 such that, for all a ≥ a0 , the metric 2 + dt 2 dst/a

on S m × S q−1 () × [0, a] has positive SGBC. In fact, via a change of variable, this is equivalent to the existence of λ0 > 0 such that, for all 0 < λ ≤ λ0 , the metric λ2 dst2 +dt 2 has positive SGBC. This is already known to be true, again by formula (7). This completes the proof of the theorem. Corollary 5.1. Let G be a finitely presented group. Then for every n ≥ 6, there exists a compact n-manifold M with positive SGBC such that π1 (M) = G. Proof. Let G be a group with presentation consisting of k generators x1 , x2 , . . . , xk , and l relations r1 , r2 , . . . , rl . Let S 1 × S n−1 be endowed with the standard product metric, which has positive SGBC (we have n − 1 ≥ 4). Note that the fundamental group of S 1 × S n−1 is infinite cyclic. Hence by taking the connected sum N of k copies of S 1 × S n−1 , we obtain an orientable compact n-manifold with positive SGBC (since this operation

310


is a surgery of codimension n ≥ 5). By the van Kampen theorem, the fundamental group of N is a free group on n generators, which we denote by x1 , x2 , . . . , xk . We now perform surgery l times on the manifold N , killing in succession the elements r1 , r2 , . . . , rl . The result is a compact, orientable n-manifold M, with positive SGBC (since the surgery is of codimension n − 1 ≥ 5), such that π1 (M) = G, as desired. References [Besse 1987] A. L. Besse, Einstein manifolds, vol. 10, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer, Berlin, 1987. MR 88f:53087 Zbl 0613.53001 [Gromov and Lawson 1980] M. Gromov and H. B. Lawson, Jr., “The classification of simply connected manifolds of positive scalar curvature”, Ann. of Math. (2) 111:3 (1980), 423–434. MR 81h: 53036 Zbl 0463.53025 [Kulkarni 1972] R. S. Kulkarni, “On the Bianchi identities”, Math. Ann. 199 (1972), 175–204. MR 49 #3767 Zbl 0234.53021 [Labbi 1997a] M.-L. Labbi, “Actions des groupes de Lie presque simples et positivité de la pcourbure”, Ann. Fac. Sci. Toulouse Math. (6) 6:2 (1997), 263–276. MR 99e:53052 Zbl 0898.53032 [Labbi 1997b] M.-L. Labbi, “Stability of the p-curvature positivity under surgeries and manifolds with positive Einstein tensor”, Ann. Global Anal. Geom. 15:4 (1997), 299–312. MR 98e:53065 Zbl 0888.53033 [Labbi 2000] M.-L. Labbi, “On compact manifolds with positive isotropic curvature”, Proc. Amer. Math. Soc. 128:5 (2000), 1467–1474. MR 2000j:53049 Zbl 0948.53017 [Labbi 2005] M.-L. Labbi, “Double forms, curvature structures and the ( p, q)-curvatures”, Trans. Amer. Math. Soc. 357:10 (2005), 3971–3992. MR 2006g:53039 Zbl 1077.53033 [Labbi 2006] M.-L. Labbi, “On two natural metrics on a tubular neighborhood of an embedded submanifold”, Preprint, 2006. math.DG/0607523 [Thorpe 1969] J. A. Thorpe, “Some remarks on the Gauss-Bonnet integral”, J. Math. Mech. 18 (1969), 779–786. MR 41 #963 Zbl 0183.50503 Received February 8, 2005. Revised June 15, 2005. M OHAMMED -L ARBI L ABBI D EPARTMENT OF M ATHEMATICS C OLLEGE OF S CIENCE U NIVERSITY OF BAHRAIN 32038 I SA T OWN BAHRAIN [email protected]


GREEN CURRENTS FOR MODULAR CYCLES IN ARITHMETIC QUOTIENTS OF COMPLEX HYPERBALLS M ASAO T SUZUKI We obtain a Green current in the sense of Gillet–Soulé on an arithmetic quotient of a complex hyperball for the modular cycle stemming from a complex subhyperball of codimension greater than one, generalizing the classical construction of the automorphic Green function for the modular curves.

Introduction and basic notation Let X be a complex manifold and Y its analytic subvariety of codimension r . The Green current for Y is defined to be a current G of (r − 1, r − 1)-type on X such that ddc G + δY is represented by a C ∞ -form of (r, r )-type on X . In the arithmetic intersection theory developed by Gillet and Soulé, the role played by the algebraic cycles in the conventional intersection theory is replaced with the arithmetic cycles. In a heuristic sense, Green currents are regarded as the “archimedean” ingredient of such arithmetic cycles [Gillet and Soulé 1990]. Consider the case when X is the quotient of a Hermitian symmetric domain G/K by an arithmetic lattice 0 in the semisimple Lie group G, and Y is a modular cycle stemming from a modular imbedding H/H ∩ K ,→ G/K , where H is a reductive subgroup of G such that H ∩ K is maximally compact in H . Inspired by the classical works on the resolvent kernel functions of the Laplacian on Riemannian surfaces [Hejhal 1983] and also by a series of works [Miatello and Wallach 1989; 1992], T. Oda posed a plan to construct a Green current for Y making use of a secondary spherical function on H \G, giving evidence for the divisorial case with some conjectures. Among many possible choices of the Green currents for a modular cycle Y , this construction may provide a way to fix a natural one. If r = 1, namely Y is a modular divisor, we already obtained a satisfactory result by properly introducing the secondary spherical functions [Oda and Tsuzuki 2003]. So it is quite natural to ask whether the same method works for the higher codimensional case. Here we focus on the case when G/K is an n-dimensional complex hyperball and H/H ∩ K is also a complex hyperball of codimension r > 1. MSC2000: 14G40, 11F72. Keywords: Green current, spherical functions, Poincaré series. 311

312

MASAO TSUZUKI

After two preliminary sections, in the third section we introduce the vectorvalued secondary spherical function φs together with its simple characterization in P Theorem 3.2. Then we form the Poincaré series G s (g) = γ ∈H ∩0\0 φs (γ g) which is shown in Theorem 4.2 to be L 1 -convergent to yield an (r − 1, r − 1)-current on 0\G/K when the parameter s lies in some right-half plane. In order to analyze G s d j further, in Section 5 we study its derivatives (− 2s1 ds ) G s with sufficiently large j by the same technique used in [Gon and Tsuzuki 2002]. Through the inductive argument of Theorem 6.2, we show in Theorem 6.3 that the function s 7→ G s has a meromorphic continuation to the whole s-plane with a functional equation and has a simple pole at the point s = n − 2r + 2. We put G to be the constant term of the Laurent expansion of G s at s = n − 2r + 2 and prove in Theorem 6.5 that the (r − 1, r − 1)-current G is a Green current for our Y . For that purpose we introduce another (r, r )-current 9s as a suitable Poincaré series and study its properties. Among other things we show that the value of 9s at its regular point s = n−2r +2 is square-integrable harmonic form representing the current ddc G+δ D up to a constant multiple; see Theorem 7.5. This paper is a continuation of the joint work of Professor Takayuki Oda and the author [Oda and Tsuzuki 2003]. Thanks are due to Professor Takayuki Oda for his interest in this work and fruitful discussions. 1. Preliminaries Let n and r be integers such that 2 6 r < n/2. For a matrix X = (xi j ) ∈ Mn (C), X ∗ denotes its conjugate transpose (x¯ ji ). Consider the two involutions σ and θ of the Lie group G = U(n, 1) := {g ∈ GLn+1 (C) | g ∗ In,1 g = In,1 } defined by θ (g) = In,1 g In,1 and σ (g) = S g S respectively. Here In,1 := diag(In , −1) and S = diag(In−r , −Ir , 1). Then K := {g ∈ G | θ(g) = g} ∼ = U(n)× U(1) is a maximal compact subgroup in G and H := {g ∈ G | σ (g) = g} ∼ = U(n − r, 1) × U(r ) is a symmetric subgroup of G such that K H := H ∩ K ∼ = U(n − r ) × U(r ) × U(1) is maximally compact in H . Pn The group G acts on the hyperball D = {z = t (z 1 , . . . , z n ) ∈ Cn | i=1 |z i |2 < 1} transitively by the fractional linear transformation g11 z + g12 g g , g = 11 12 ∈ G, z ∈ Cn . g·z = g21 g22 g21 z + g22 Under the identification G/K ∼ = D of G-manifolds sending gK to g·0, the quotient H/K H corresponds to the H -orbit of 0, i.e., D H := {z ∈ D | z n−r +1 = · · · = z n = 0}. In particular the real codimension of H/K H in G/K is 2r . The Lie algebra g := Lie(G) is realized in its complexification gC = gln+1 (C) as an R-subalgebra of all X ∈ gln+1 (C) such that X ∗ In,1 + In,1 X = On+1 . Let p

GREEN CURRENTS FOR MODULAR CYCLES

313

be the orthogonal complement of k := Lie(K ) in g with respect to the G-invariant, nondegenerate R-bilinear form hX, Y i = 2−1 tr(X Y ) on g. We have the Cartan decomposition g = k ⊕ p. We put G/K a unique G-invariant complex structure such that the map G/K ∼ = D is biholomorphic. For 1 6 i, j 6 n + 1, let Ei, j := (δui δv j )uv ∈ gln+1 (C) be the matrix unit. Put X i := Ei,n+1 and X i = En+1,i ,

1 6 i 6 n − 1;

X 0 := En,n+1 and X 0 = En+1,n . Pn−1 Then p+ = i=0 C X corresponds to the holomorphic tangent space of G/K at Pn−1 i K , and p− = i=0 C X i to the antiholomorphic tangent space. V The exterior algebra p∗C is decomposed to the direct sum of subspaces V p,q ∗ Vp ∗ Vq ∗ pC := p+ ⊗ p− , p, q ∈ N. With {ωi } the basis of p∗+ dual to {X i } and {ωi } the basis of p∗− dual to {X i }, put ω :=

√ −1 2

n−1 X

ωi ∧ ωi ∈

V1,1

p∗C ∩

p∗

V

and

vol :=

1 n n! ω

∈

Vn,n

p∗C ∩

V

p∗ .

i=0

The dual inner product on p∗ naturally extends to the Hermitian inner product ( · | · ) V V of p∗C . The Hodge star operator ∗ is the C-linear automorphism of p∗C such that V ∗ ∗α = ∗α and (α|β) vol = α ∧ ∗β, for α, β ∈ pC . We remark that √ {X j + X j , −1(X j − X j )}n−1 j=0 √ is an orthonormal basis of p, dual to {2−1 (ω j + ω j ), −2−1 −1(ω j − ω j )}n−1 j=0 . For V ∗ V ∗ V ∗ α ∈ pC , define the endomorphism e(α) : pC → pC by e(α)β = α ∧ β. As usual, we have the Lefschetz operator L := e(ω) and its adjoint operator 3 = e∗ (ω) V acting on the finite dimensional Hilbert space p∗C [Wells 1980, Chapter V]. Put h = Lie(H ). Then θ restricts to a Cartan involution of h giving the decomposition h = (h ∩ k) ⊕ (h ∩ p). The complex structure of p induces that of h ∩ p giving the decomposition (h ∩ p)C = (h ∩ p)+ ⊕ (h ∩ p)− with (h ∩ p)+ = hC ∩ p+ =

n−r X

and

CXi

(h ∩ p)− = hC ∩ p− =

i=1

n−r X

CXi .

i=1

We introduce two tensors ω H and η as ω H :=

√ −1 2

n−r X i=1

ωi ∧ ωi

and η :=

n−1 X −1 ωi 2 j=n−r +1

√

∧ ωi = ω − ω H −

√

−1 2 ω0 ∧ ω0 .

314

MASAO TSUZUKI

For 0 6 p 6 r − 1 let S p denote the set of all subsets J ⊂ {n − r + 1, . . . , n − 1} such that ](J ) = p. Then a computation shows that √ XY η p = ( 2−1 ) p p! ω j ∧ ω j , 0 6 p 6 r − 1. J ∈S p j∈J

From [Wells 1980, (1.5), p. 163], we have 3(η p ) = p(r − p) η p−1 ,

(1)

0 6 p 6 r − 1.

The coadjoint representation of K on p∗ is extended to the unitary representation V τ : K → GL( p∗C ) in such a way that τ (k)(α ∧ β) = τ (k)α ∧ τ (k)β holds for all V ∗ α, β ∈ pC and k ∈ K . The differential of τ is also denoted by τ . Then we have τ (Z )(α ∧ β) = (τ (Z )α) ∧ β + α ∧ (τ (Z )β),

(2)

(τ (Z )α|β) = −(α|τ (Z )β) for α, β ∈ p∗C , Z ∈ kC . V The irreducible decomposition of the K -invariant subspaces p,q p∗C is wellknown. V

Lemma 1.1. Let p, q be nonnegative integers such that p + q 6 n. Put V F p,q := α ∈ p,q p∗C | 3(α) = 0 . Then F p,q is an irreducible K -invariant subspace of p∗C . The K -homomorphism V V L induces a linear injection p−1,q−1 p∗C → p,q p∗C whose image is the orthogonal V p,q ∗ complement of F p,q in pC . In other words, V p,q ∗ V p−1,q−1 ∗ pC = F p,q ⊕ L pC . V

Proof. Use [Borel and Wallach 1980, Lemma 4.9, p. 199].

The one dimensional R-subspace a = RY0 with Y0 := X 0 + X 0 ∈ p is a maximal abelian subalgebra in q ∩ p with q the (−1)-eigenspace of dσ , the differential of σ . Since (G, H ) is a symmetric pair, by the general theory [Heckman and Schlichtkrull 1994, Theorem 2.4, p. 108], the group G is a union of double cosets H at K (t > 0) with cosh t sinh t at := exp(tY0 ) = diag In−1 , , t ∈ R. sinh t cosh t Put A = {at | t ∈ R}. Let M0 be the group of all the elements k ∈ K such that Ad(k)Y0 = Y0 and set M = M0 ∩ H . Then M0 = {diag(u, u 0 , u 0 ) | u ∈ U(n − 1), u 0 ∈ U(1)}, M = {diag(u 1 , u 2 , u 0 , u 0 ) | u 1 ∈ U(n − r ), u 2 ∈ U(r − 1), u 0 ∈ U(1)}.


315

It is important to know the structure of the M-fixed part of F p, p . Proposition 1.2. Let p be an integer such that 0 < p 6 r . Put ( p) v0

p

√ X ( p) 1 = c j L p− j (n− p− j+1)η j + 2−1 j (r − j) ω0 ∧ ω0 ∧ η j−1 , n − p+1 j=0

( p) v1

p

√ X ( p) −1 = c j L p− j ( p− j)η j + 2−1 j (r − j) ω0 ∧ ω0 ∧ η j−1 , p(n −2 p+1) j=0

where ( p)

c j = (−1) j

p n− p+1 r −1−1 ,0 j j j

6 j 6 inf( p, r −1), ( p)

and

cr(r ) = 1.

( p)

M is a two-dimensional space generated by v Then F p, p 0 and v1 .

Proof. Let W be the C-span of elements ωi , ωi (1 6 i 6 n −1). Then for each p, q, V p,q ∗ V V pC is an orthogonal direct sum of four subspaces p,q W , e(ω0 )( p−1,q W ), V V V e(ω0 )( p,q−1 W ) and e(ω0 ∧ ω0 )( p−1,q−1 W ). The space W has the Lefschetz V operator L 0 and its adjoint 30 . These are also given as L 0 = e(ω H + η)| W V and 30 = 3| W . Let p, q be natural numbers such that p + q < n. Then the V V subspace E p,q = {β ∈ p,q W | 30 (β) = 0} is an irreducible M0 -subspace of W ; see Lemma 1.1. First we show that the linear map T p,q from the direct sum E p,q ⊕ V E p−1,q ⊕ E p,q−1 ⊕ E p−1,q−1 to p∗C defined by sending (β1 , β2 , β3 , ξ ) to T p,q (β1 , β2 , β3 , ξ )

= β1 + (β2 ∧ ω0 ) + (β3 ∧ ω0 ) +

√ −1 2 ξ

∧ ω0 ∧ ω 0 −

L 0 (ξ ) n − p−q +1

is an M0 -isomorphism onto F p,q . V An element ζ ∈ p,q p∗C is expressed uniquely using an element (ζ1 , β2 , β3 , ζ4 ) V V V V in the direct sum p,q W ⊕ p−1,q W ⊕ p,q−1 W ⊕ p−1,q−1 W as the sum ζ = ζ1 + (β2 ∧ ω0 ) + (β3 ∧ ω0 ) + (ζ4 ∧ ω0 ∧ ω0 ). We examine the condition 3(ζ ) = 0. Since the four equalities 3(ζ1 ) = 30 (ζ1 ), 3(β2 ∧ω0 ) = 30 (β2 )∧ω0 , 3(β3 ∧ω0 ) = 30 (β3 ) ∧ ω0 and 3(ζ4 ∧ ω0 ∧ ω0 ) = 30 (ζ4 ) ∧ ω0 ∧ ω0 + √2−1 ζ4 hold, the √ condition 3(ζ ) = 0 is equivalent to 30 (β2 ) = 30 (β3 ) = 30 (ζ4 ) = 0 and ζ4 = − 2−1 30 (ζ1 ). V We can write ζ1 = β1 −(n − p −q +1)−1 L 0 (ξ ) with β1 ∈ E p,q and ξ ∈ p−1,q−1 W uniquely; see Lemma 1.1. By [Wells 1980,√ Proposition 1.1(c), p. 160], we have 30 L 0 (ξ ) = (n − p−q +1)ξ and hence ζ4 = 2−1 ξ . Since 30 (ζ4 ) = 0, ξ ∈ E p−1,q−1 . Consequently we arrive at ζ = T p,q (β1 , β2 , β3 , ξ ) to know the image of T p,q is F p,q . The injectivity of T p,q follows from Im L 0 ∩ E p,q = {0}. Since ω0 , ω0 are M0 -invariant and since L 0 is an M0 -homomorphism, the map T p,q is also an M0 -homomorphism.

316

MASAO TSUZUKI

The map T p, p induces a linear isomorphism M M M M ∼ M E p, p ⊕ E p−1, p ⊕ E p, p−1 ⊕ E p−1, p−1 = F p, p

(3)

of the M-invariant part. M = {0} if p 6 = q, p + q < n. For 0 6 p 6 r , E M = C u( p) 6 = {0}, where Claim: E p,q p, p

u( p) =

p X n − p − j + 1 ( p) p− j j c j L 0 (η ). n− p+1 j=0

Granting this, we easily prove the proposition easily. Indeed, a direct computa( p) ( p) tion yields v0 = T p, p (u( p) , 0, 0, 0) and v1 = T p, p (0, 0, 0, u( p−1) ). The claim, M . together with the decomposition (3) implies these tensors form a basis of F p, p We prove the claim. The element diag(In−1 , u 0 , u 0 ) (u 0 ∈ U(1)) of M acts on − p+q M = {0} if p 6 = q. E p,q by the scalar u 0 . Hence E p,q From now on, let 0 6 p 6 r . Since M is a symmetric subgroup of M0 and M ) 6 1. Hence to show that E p, p is an irreducible M0 -module, we have dimC (E p, p M = C u( p) 6 = {0}, it is sufficient to prove that the element u( p) is M-invariant, E p, p nonzero and 30 (u( p) ) = 0. Since η is M-invariant and L 0 is an M0 -homomorphism, the M-invariance of u( p) is clear. We prove that it is nonzero using the expression p X α X n − p − j + 1 ( p) p − j p−α ( p) (4) ω H ∧ ηα cj u = n− p+1 α− j α=0

j=0

which we get from the original formula for u( p) by substituting the binomial exp− j p−α pansion of L 0 = e((ω H + η) p− j ). It is easy to see that the tensors ω H ∧ ηα p ( p) are linearly independent. The coefficient of ω H in expression (4) is c0 , which is nonzero. Hence we conclude u( p) 6= 0. There remains to show that 30 (u( p) ) = 0. By [Wells 1980, Proposition 1.1(c), p. 160] and (1), we have p− j

30 L 0

p− j

(η j ) = L 0

p− j−1

30 (η j ) + ( p − j)(n − j − p) L 0 p− j

= j (r − j) L 0

(η j ) p− j−1

(η j−1 ) + ( p − j)(n − j − p) L 0

(η j ).

Hence 30 (u( p) ) equals (n − p + 1)−1 times p−1 X

( p)

( p)

(n− p− j) ( j+1)(r − j−1) c j+1 + (n− p− j+1)( p− j) c j

p− j−1

L0

(η j ).

j=0 p− j−1

We can easily check that the coefficient of each L 0 is zero.

(η j ) in the expression above


317

For convenience, we put v0(0) = 1, v1(0) = 0; these are elements of F0,0 = C. We also need the structure of the (H ∩ K )-invariant part of F p, p (0 6 p < r ). Proposition 1.3. For each 0 < p 6 r − 1, put w

( p)

=

p X

( p)

(r − j)c j L p− j ((ω − ω H ) j ).

j=0

For p = 0, we set w(0) = r . Then (5)

w( p) =

(r − p)(n − p + 1) ( p) ( p) v0 − p(n − p − r + 1) v1 , n − 2p + 1

H ∩K = C w( p) 6 = {0}. and F p, p

Proof. The (H ∩K )-invariance of w( p) follows from the fact that ω−ω H is (H ∩K )invariant and L is a K -homomorphism. The formula (5) follows from a direct computation. Since H ∩ K = U(n − r ) × U(r ) × U(1) is a symmetric subgroup of H ∩K ) 6 1. Since w( p) 6 = 0 belongs to F H ∩K , the K = U(n) × U(1), we have dim(F p, p p, p last assertion follows. For convenience, we put µ = r − 1 and l = n − 2r + 2. Then our assumption 2 6 r < n/2 is equivalent to µ > 1 and λ > 3. 1 Proposition 1.4. Define w := (ω − ω H )µ and vol H = ωn−r . We have (n −r )! H √ 1 (6) ∗vol H = w ∧ (ω − ω H ), w ∧ (η − 2−1 µ ω0 ∧ ω0 ) = 0, r! µ X 1 3(∗vol H ) = (7) w, w= γ p L µ− p (w( p) ), µ! p=0

where the γ p , 0 6 p 6 µ, are real numbers defined by the recurrence relation γµ =

1 (µ) cµ

,

( j) γj cj

=−

µ X

( p)

γp c j ,

0 6 j < µ.

p= j+1

Proof. Using [Wells 1980, Lemma 1.2, p. 161], we easily check the first formula r µ in √ (6) by a computation. The second follows directly from η = 0 and w = η + −1 µ−1 . The first formula in (7) is a consequence of (6) and [Wells 2 µ ω0 ∧ ω0 ∧ η 1980, (1.5), p. 163]. To justify the last formula, note that its right-hand side equals µ X µ X ( p) γ p c j (r − j)L µ− j (ω − ω H ) j ; j=0

p= j

this equals w by the definition of w and γ j .

318

MASAO TSUZUKI

2. Radial parts of several differential operators We define 2n − 1 elements Z α , Z α and Z 0 of kC by Z α = Eα,n , Z α = −En,α , 1 6 α 6 n − 1, √ Z 0 = −1(En,n − En+1,n+1 ). Note that (k∩h)C is a direct sum of Lie(M)C and the C-span of the 2r −1 elements Z j , Z j (n − r + 1 6 j 6 n − 1), Z 0 . Lemma 2.1. For 1 6 α, β 6 n−1, we have

(8)

τ (Z α )ωβ = 0,

τ (Z α )ωβ = −δαβ ω0 ,

τ (Z α )ω0 = ωα , √ τ (Z 0 )ωα = − −1ωα ,

τ (Z α )ω0 = 0,

√ τ (Z 0 )ω0 = −2 −1ω0 .

For n − r + 1 6 α 6 n − 1, we have τ (Z α )η = −

√

−1 2 ω0 ∧ ωα ,

τ (Z α )η = −

√

−1 2 ωα

∧ ω0 .

Definition. Let Cτ∞ be the space of C ∞ -functions ϕ : G − H K → p∗C having the (H, K )-equivariance property ϕ(hgk) = τ (k)−1 ϕ(g) for h ∈ H , k ∈ K . V

Lemma 2.2. Let i, ν ∈ {1, . . . , n −r } and j, µ ∈ {n −r +1, . . . , n −1}. Let ϕ ∈ Cτ∞ . Then: d δ (9) R X i X ν ϕ(at ) = iν tanh t + τ (Z i Z ν ) tanh 2 t 2 dt √ −1 + δiν (1 + tanh 2 t)τ (Z 0 ) ϕ(at ) 4 √ τ (Z i ) d −1 tanh t + (tanh 2 t + 1)τ (Z 0 ) ϕ(at ) R X i X 0 ϕ(at ) = 2 dt 2 √ τ (Z i ) d −1 R X 0 X i ϕ(at ) = tanh t + 2 − (tanh 2 t + 1)τ (Z 0 ) ϕ(at ) 2 dt 2 √ 1 d2 1 d −1 + (tanh t + coth t) + τ (Z 0 ) (10) R X 0 X 0 ϕ(at ) = 4 dt 2 4 dt 2 1 + (tanh t + coth t)2 τ (Z 0 )2 ϕ(at ) 16 R X i X j ϕ(at ) = τ (Z i Z j )ϕ(at ) R X j X i ϕ(at ) = τ (Z j Z i )ϕ(at )


δ

319

d + τ (Z j Z µ ) coth2 t 2 dt √ −1 + (1 + coth2 t)δ jµ τ (Z 0 ) ϕ(at ), 4 √ τ (Z j ) −1 d R X j X 0 ϕ(at ) = coth t + (1 + coth2 t)τ (Z 0 ) ϕ(at ), 2 dt 2 √ τ (Z j ) d −1 coth t + 2 − (1 + coth2 t)τ (Z 0 ) ϕ(at ). R X 0 X j ϕ(at ) = 2 dt 2 Proof. We prove these formulas through a computation similar to that of [Oda and Tsuzuki 2003, Lemma 7.1.2], using the formulas (11) R X j X µ ϕ(at ) =

jµ

coth t

1 Ad(at )−1 X i , 1 6 i 6 n − r, cosh t 1 X j = − coth t · Z j + Ad(at )−1 Z j , n − r + 1 6 j 6 n − 1, sinh t √ √ −1 1 −1 Ad(at )−1 Z 0 X 0 = Y0 + coth(2t) Z0 − 2 2 2 sinh (2t) X i = − tanh t · Z i +

and their complex conjugates.

Casimir operators. Let , K and M0 be the Casimir elements of g, k and m0 := Lie(M0 ) corresponding to the invariant form hX, Y i = 2−1 tr(X Y ). Since X α +X α , √ n−1 −1(X α − X α ) α=0 is an orthonormal basis of p, we have = K + 2

(12)

n−1 X

(X α X α + X α X α ).

α=0

√ n−1 Since Z β + Z β , −1(Z β − Z β ) β=1 ∪ {Z 0 } is a pseudo-orthonormal basis of the orthogonal complement of m0 in k, we have

K = M0 − 2

(13)

n−1 X (Z β Z β + Z β Z β ) − Z 02 . β=1

Theorem 2.3. Let ϕ ∈ Cτ∞ be such that its values ϕ(g), g ∈ G, belong to p, p p∗C for 0 6 p 6 r . Put f (z) = ϕ(at ) where z = tanh 2 t. Then ϕ(at ) (t > 0) equals 2 S S0 τ ( M0 ) d µ+1 n−1 d 2 + + − + + f (z), 4z(1−z) dz 2 z 1−z dz z(1−z) z 2 (1−z) 4z(1−z)2 V

where S and S0 are operators on S := 2−1

n−r X i=1

V

p∗C defined by

τ (Z i Z i + Z i Z i ) and

S0 := 2−1

n−1 X j=n−r +1

τ (Z j Z j + Z j Z j ).

320

MASAO TSUZUKI

Proof. By (8) and Proposition 1.2, the operator τ (Z 0 ) is zero on the space

V ( j) ( j) ( p, p p∗C ) M = L p− j v0 , L p− j v1 | 0 6 j 6 p C , to which the values ϕ(at ) (t > 0) belong by the (H, K )-equivariance of ϕ. Hence under the assumption, we have τ (Z 0 )ϕ(at ) = 0 identically. We also note that the V center of K acts on p, p p∗C trivially. Noting these remarks and using the fact that Pn−1 α=0 [X α , X α ] is in the center of kC , we obtain the following expression of ϕ(at ) from (12) and (13). ϕ(at ) = 4

n−1 X α=0

R X α X α ϕ(at ) − 2

n−1 X

τ (Z β Z β + Z β Z β )ϕ(at ) + τ ( M0 )ϕ(at ).

β=0

We can compute the first n terms of the right-hand side of this equation using the formulas (9), (10) and (11) to obtain 2 4S d d 2r −1 + (2n−2r +1) tanh t + − (14) ϕ(at ) = 2 tanh t dt cosh 2 t dt 4S0 + + τ ( M0 ) ϕ(at ). sinh 2 t The conclusion follows since 2 d2 2 d + 2(1−z)(1−3z) d , = 4z(1−z) dt 2 dz 2 dz

(15)

d d = 2z 1/2 (1−z) . dt dz

Lemma 2.4. Let 0 < p 6 µ and put l = n − p − r + 1 and m = r − p. Then: (16)

1 l +m −1 ( p) l ( p) ( p) ( p) Sv = − Sv = v + v1 , mp 0 m(l + p + m) 1 l +m 0 m

(17) −

1 0 ( p) l +m 1 ( p) ( p) ( p) S v0 = − S0 v1 = v0 + v1 , lp m(l + p + m) l +m ( p)

( p)

( p)

( p)

(18) τ ( M0 )v0 = 4 p(n − p) v0 and τ ( M0 )v1 = 4( p − 1)(n − p + 1) v1 . ( p)

( p)

Proof. As shown in the proof of Proposition 1.2, the tensors v0 and v1 are obtained as the image of u( p) ∈ E p, p and u( p−1) ∈ E p−1, p−1 respectively by the M0 homomorphism T p, p . Hence in order to prove (18) we have only to check that the eigenvalue of M0 on E p, p equals 4 p(n − p). The eigenvalue is easily calculated if we note that the highest weight of an irreducible M0 = (U(n −1)× U(1))-module E p, p is p

n−2 p−1

p

z }| { z }| { z }| { (1, . . . , 1, 0, . . . , 0, −1 . . . , −1; 0). We prove the formula (17). Since w( p) is (H ∩ K )-invariant and since the elements Z j , Z j (n − r + 1 6 j 6 n − 1) belong to (h ∩ k)C , we have S0 w( p) = 0. This


321

together with (5) gives the first equality of (17). To obtain the second equality we need some computation. From definition we have the expressions √ −1 c(pp) 2−1 (r − p) ω0 ∧ ω0 ∧ η p−1 + L(ξ1 ), n −2 p+1 (19) √ 1 ( p) v0 = c(pp) (n−2 p+1)η p + 2−1 p(r − p) ω0 ∧ ω0 ∧ η p−1 + L(ξ0 ), n − p+1 ( p)

v1 =

for some ξ0 and ξ1 . Using Lemma 2.1 and (2), we compute that τ (Z j Z j )(ω0 ∧ ω0 ∧ η p−1 ) = ω j ∧ ω j ∧ η p−1 + ω0 ∧ (−ω0 ) ∧ η p−1 −

√ −1 2 ( p − 1) ω0 ∧ ω j

∧ ω j ∧ ω0 ∧ η p−2

for n − r + 1 6 j 6 n − 1. Taking the sum over j, we obtain n−1 X

τ (Z j Z j )(ω0 ∧ ω0 ∧ η p−1 ) = −(r − p) ω0 ∧ ω0 ∧ η p−1 + √2−1 η p .

j=n−r +1 ( p)

( p)

We use (19) to write the right-hand side in terms of v0 and v1 : n−1 X

τ (Z j Z j )(ω0 ∧ ω0 ∧ η p−1 )

j=n−r +1

2 n− p+1 =√ ( p) −1 cp

1 ( p) ( p) + L(ξ2 ) v + v1 n − 2p + 1 0

for some ξ2 . From this and the first equality in (19), we obtain (20)

n−1 X

τ (Z

( p) j Z j )v1

j=n−r +1

(r − p)(n− p+1) =− n − 2p + 1

1 ( p) ( p) + L(ξ3 ) v + v1 n−2 p+1 0

for some ξ3 . Since all terms in this identity except L(ξ3 ) belong to F p, p and since ( p) ( p) F p, p ∩ Im L = {0}, the residual term L(ξ3 ) has to be zero. Noting that v0 and v1 V ∗ ( p) are real elements of pC , we know that 2S0 v1 is given by the sum of (20) and its complex conjugate. This completes the proof of the second identity of (17). We can deduce (16) from (17) and (18) using the relation ( p)

τ ( K )vi

( p)

= τ ( M0 )vi

( p)

− 4Svi

( p)

− 4S0 vi

obtained from (13), since the eigenvalue of K on F p, p equals 4 p(n + 1 − p). Let 0 < p 6 µ and ϕ ∈ Cτ∞ be such that its values ϕ(g) belong to F p, p . By the (H, K )-equivariance, the vector ϕ(at ) belongs to the space of M-fixed tensors M = C v( p) ⊕ C v( p) . We can write F p, p 0 1 (21)

( p)

( p)

( p) ( p)

ϕ(at ) = f 0 (z) v0 + f 1 (z) v1 = [v0 v1 ]F(z),

z = tanh 2 t ∈ (0, 1),

322

MASAO TSUZUKI

with a C2 -valued C ∞ -function f 0 (z) F(z) = f 1 (z)

on 0 < z < 1. Given a complex number s, consider the differential equation ϕ(g) = (s 2 − λ2 )ϕ(g),

(22)

g ∈ G − H K.

We rewrite this equation in terms of F(z). " # 0 1 Q (z) Q (z) 0 0 Proposition 2.5. Let Q( p) (z) := with Q 01 (z) Q 11 (z) Q 00 (z)

1 pm 2 lp 2 1 2 = 2 − z + p − (m−1)l + 4 (n − s ) z − , z (1 − z)2 l +m l +m

Q 10 (z) = −

m(l + p + m) 1 + z , (l + m)2 z 2 (1 − z)

Q 01 (z) = −lp Q 11 (z)

1+z z 2 (1 − z)

,

l(l + p + m) 2 1 2 1 2 + p − (m − 1)l + − s ) z − z (n = 2 4 z (1 − z)2 l +m m(l + p + m) − . l +m

Then ϕ ∈ Cτ∞ with values in F p, p is a solution of (22) if and only if F(z) (0 < z < 1) satisfies the second order ordinary differential equation µ + 1 n − 1 dF d 2F + + + Q( p) (z)F = 0. dz 2 z 1 − z dz

(23)

Proof. This follows from Theorem 2.3 and Lemma 2.4. The Schmid operator. Let ∇± be the Schmid operator, which is given by ∇+ f (g) =

n−1 X i=0

e(ωi )R X i f (g) and

∇− f (g) =

n−1 X

e(ωi )R X i f (g)

i=0

for a p∗C -valued C ∞ -function f on an open subset of G. To describe the radial V part of the Schmid operator, we introduce four operators acting on p∗C : V

GREEN CURRENTS FOR MODULAR CYCLES n−r X

P+ =

R+ =

e(ωi )τ (Z i ),

P− =

n−r X

i=1

i=1

n−1 X

n−1 X

e(ω j )τ (Z j ),

R− =

323

e(ωi )τ (Z i ), e(ω j )τ (Z j ).

j=n−r +1

j=n−r +1

Theorem 2.6. Let ϕ ∈ Cτ∞ and f (z) be the same as in Theorem 2.3. Then for t > 0, ∇− ∇+ ϕ(at ) equals A B d d2 2 + (24) z(1−z) e(ω0 ∧ ω0 ) 2 − dz z 1−z dz P− P+ R− R+ C + + + z f (z) (1−z)2 z 2 (1−z)2 z(1−z)2 where the numerators A, B and C are given by A = − √2 e(ω − ω H ) + e(ω0 )R− − e(ω0 )R+ , B=

−1 √ − 2−1 e(ω) + e(ω0 ∧ ω0 ) + e(ω0 )(P− + R− ) − e(ω0 )(P+ + R+ ),

C = e(ω0 )(P+ + R+ ) + P− R+ + R− P+ .

Proof. Input the formulas in Lemma 2.2 to the right-hand side of the identity ∇− ∇+ ϕ(at ) =

n−1 X α,β=0

R X α X β ϕ(at ) ∧ ωα ∧ ωβ .

Then a direct computation using (15) yields (24).

3. Secondary spherical functions Before we state the main theorem of this section, we prove a lemma which is important not only in this section but also in the global theory to be developed in Section 5. Lemma 3.1. For each integer p with 1 6 p 6 r , there exists a unique holomorphic ( p) function s 7→ νs on the domain C − L p with p √ (25) L p = s ∈ −1 R | 2 (r − p)(n − p − r + 2) > |Im s| , which takes a positive real value for s > 0 and such that {νs( p) }2 = s 2 + 4(r − p)(n − p − r + 2). ( p)

( p)

The functional equation ν−s = −νs ( p) ( p+1) Re νs > Re νs > Re s.

holds for s ∈ C − L p . Also, if Re s > 0, then

324

MASAO TSUZUKI

Proof. √ Put c p = 4(r − p)(n − p − r + 2). By the residue theorem, the value (mod 2π −1Z) of the integral Z ζ I (s) := dζ 2 Cs ζ + c p is independent of the choice of the path Cs connecting 1 and s inside the region ( p) C − L p , and locally defines a holomorphic function on C − L p . Since νs = p exp(I (s)) c p + 1 for s ∈ C − L p , the first assertion follows. Put σ = Re s and t = Im s and suppose σ > 0. Then we have p 1/2 Re νs( p) = 2−1/2 x p,s + x 2p,s + ys2 , with x p,s = σ 2 − t 2 + c p , ys = 2σ t. Since c p > c p+1 and x p,s > x p+1,s , we have √ 1/2 ( p) Re νs( p) > Re νs( p+1) . The formula Re νs = σ 2 + 2−1 ( T 2 + 4c p σ 2 − T ) ( p) with T = t 2 + σ 2 − c p obviously implies the inequality Re νs > σ . ( p) ( p) To prove ν−s = −νs , take a path Cs,−s lying in the domain C − L p from s to −s and C 0 the image of Cs,−s by the map z = ζ 2 + c p . Then C 0 is a simple loop ( p) ( p) rounding the point ζ = 0. By definition, νs = exp(I (s) − I (−s))ν−s . Since Z Z √ ζ 1 dz I (s) − I (−s) = −1, dζ = = ±π 2 2 C0 z Cs,−s ζ + c p ( p)

( p)

we have ν−s = −νs .

Consider the holomorphic function d(s) :=

r Y

0(νs( p) )−1 0(2−1 (νs( p) − λ) + 1)−1 ,

s ∈ C − L 1,

p=1

and define the related sets D = {s ∈ C − L 1 | d(s) 6= 0},

D˜ =

µ T

( p)

{s ∈ D | Re νs

( p+1)

+ Re νs

> 4}.

p=1

˜ Note that {s ∈ C | Re s > λ} ⊂ D. The aim of this section is to prove the following theorem. Theorem 3.2. There is a unique family of C ∞ -functions φs : G − H K → ˜ satisfying the following conditions. varying over D, (i) For each g ∈ G − H K , the function s 7→ φs (g) is holomorphic. (ii) φs has the (H, K )-equivariance property φs (hgk) = τ (k)−1 φs (g),

h ∈ H, k ∈ K , g ∈ G − H K .

Vµ,µ

p∗C , s


325

(iii) φs satisfies the differential equation (26)

φs (g) = (s 2 − λ2 )φs (g),

(iv) (Recall the (H ∩ K )-invariant tensor w ∈ We have

g ∈ G − H K. Vµ,µ

p∗C defined in Proposition 1.4.)

lim t 2µ φs (at ) = w.

t→+0

(v) If Re s > n, then φs (at ) decays exponentially as t → +∞. We call the function φs the secondary spherical function. Differential equations. In this subsection, we fix an s ∈ D. Lemma 1.1 yields the decomposition (27)

Vµ,µ

p∗C =

µ X

L µ− p (F p, p )

p=0

of K -modules. Hence in order to obtain the function φs as in Theorem 3.2, we ( p) have only to construct a function φs : G − H K → F p, p for each 0 6 p 6 µ with the same properties as φs listed in Theorem 3.2 except condition (iv). Instead we require lim t 2µ φs( p) (at ) = w( p) ,

t→+0

Pµ ( p) and then form φs (g) = p=0 γ p L µ− p (φs (g)). From now on we fix 0 6 p 6 µ and examine the conditions to be satisfied by ( p) φs . We use the notation l = n − p − r + 1, m = r − p as in Lemma 2.4. ( p)

The case p > 0. Since φs should be a solution of (22), we first analyze the differential equation (23) in some detail. h i f (z) Proposition 3.3. Consider a C2 -valued C ∞ -function F(z) = 0 , 0 < z < 1. f 1 (z) The following conditions on F(z) are equivalent. (a) F(z) is a solution of (23) and also satisfies z d m(l + p + m) 1 l + p + m (28) f 0 (z) = − + + f 1 (z). lp dz l +m z 1−z (b) F(z) is a solution of (29)

dF = B( p) (z)F dz

326

MASAO TSUZUKI

h B 0 (z) B 1 (z)i 0 with entries where B( p) (z) = 00 B1 (z) B11 (z) B00 (z) = −

lp 1 p − , l +m z 1−z

B01 (z) = −

m(l + p+m) 1 s 2 −(l −m +1)2 1 , − 4lp (l +m)2 z (1−z)2

B11 (z) = −

m(l + p+m) 1 l + p+m − . l +m z 1−z

B10 (z) = −

lp , z

(c) The entries f 0 and f 1 of F(z) satisfy (28) and 2 µ + 2 n+1 d d + (30) + dz 2 z 1−z dz (n + 2)2 − 4l(m−1) − s 2 1 1 1 f 1 (z) = 0. + + + 4 z 1−z (1−z)2

Proof. This follows by direct computation.

Remark. When F(z) is related to some ϕ ∈ Cτ∞ by the relation (21), the equation (28) comes from Pr(∇+ ϕ(at )) = 0, where Pr : F p, p ⊗ p∗− → F p, p−1 is a certain K -projector. Lemma 3.4. Let F0 (z) and F1 (z) be two solutions of (29) and form the 2×2-matrix valued function 8(z) := [F0 (z) F1 (z)]. Consider a C2 -valued C ∞ -function A(z) on 0 < z < 1. Then the following two conditions on A(z) are equivalent. (a) The function F˜ (z) = 8(z) A(z) is a solution of the differential equation (23). dA (b) The function U(z) = 8(z) (z) satisfies the differential equation dz h i dU µ+1 n −1 1 0 ( p) (31) = − B (z) + + U. dz z 1−z 0 1 Proof. Put p(z) = (µ + 1)z −1 + (n − 1)(1 − z)−1 . Since 8(z) satisfies d 28 d8 + Q( p) (z)8 = 0 + p(z) 2 dz dz

and

d8 = B( p) (z)8, dz

we have 2 d d ( p) + p(z) + Q (z) (8(z)A(z)) dz 2 dz h i d ( p) d A(z) 1 0 = + B (z)+ p(z) 8(z) , 0 1 dz dz by a direct computation. This identity proves the equivalence of (a) and (b).


327

i u (z) Lemma 3.5. Consider a C2 -valued C ∞ -function U(z) = 0 on 0 < z < 1. u 1 (z) Then U is a solution of (31) if and only if it satisfies z d lp 1 p − 2 + + u 1 (z) = u 0 (z) lp dz l + m z 1 − z and 2 µ+2 n −3 d n 2 −s 2 −4l(m −1)−8(m +1) 1 1 d + + + + z 1−z dz 4 z 1−z dz 2 n 2 −s 2 −4l(m −1)−8(n − p) 1 u 1 (z) = 0. + 4 (1−z)2 h

Proof. This is a direct computation.

The case p = 0. Since the function φs(0) is F0,0 = C-valued, we can write, thanks to (H, K )-equivariance, φs(0) (at ) = f 0 (z),

z = tanh 2 t,

with a C ∞ -function f 0 (z) on 0 < z < 1. From (22) we obtain: Proposition 3.6. The function f 0 (z) satisfies the differential equation 2 d µ+1 n−1 d 1 λ2 − s 2 (32) + + + f 0 (z) = 0. dz 2 z 1 − z dz z(1 − z)2 4 Construction of solutions. Let N be a nonnegative integer and γi (0 6 i 6 N ) complex numbers. For a function f (s, z) on {(s, z) | s ∈ D, 0 < z < 1}, if there exist a meromorphic function q(s) on C − L 1 holomorphic on D and a family of holomorphic functions α(s), ai (s ; z) (0 6 i 6 N ) on {(s, z) | s ∈ C − L 1 , |z| < ε} with some ε > 0 such that N X γi i α(s) f (s ; z) = q(s)z 1+ z (log z) ai (s ; z) , 0 < z < ε, s ∈ D, i=0

then we write f (s ; z) = q(s)z α(s) (1+ O(s ; z γ0 , z γ1 log z, . . . , z γ N (log z) N )). Given a C2 -valued function f 0 (s ; z) F(s ; z) = f 1 (s ; z) on {(s, z) | s ∈ D, 0 < z < 1}, we write q0 (s)z α0 (s) γ0 γ1 γN N F(s ; z) = 1 + O(s ; z , z log z, . . . , z (log z) ) α (s) q1 (s)z 1 when f j (s ; z) = q j (s)y α j (s) 1+ O(s ; z γ0 , z γ1 log z, . . . , z γ N (log z) N ) for j = 0, 1.

328

MASAO TSUZUKI

We define three new functions: 0(s + 1) 0(µ + 2) c(s) := , 0((s + n)/2 + 1) 0((s − λ)/2 + 1) s −n s +λ h s (z) := 2 F1 − + 1, − + 1; µ + 2; z , 2 2 s −n s +λ Hs (z) := 2 F1 , ; s + 1; 1 − z . 2 2 Note that c(s) has no poles nor zeros in the domain D. Lemma 3.7. Let s ∈ D. Then h s (z) = 1 + O(s ; z) and

Hs (z) =

c(s) 1 + O(s ; z, z µ+1 log z) . µ+1

With y = 1 − z, we have c(s) (1 + O(s ; y)) if Re s > 0, and Hs (z) = 1 + O(s ; y). s Proof. Use [Magnus et al. 1966, p. 49, line 6 and p. 47, last line]. h s (z) =

The case p > 0. Proposition 3.8. For i = 0, 1, define by ( p+1)

( p)

f 10 (s ; z) = (1−z)(−νs ( p) f 11 (s ; z)

+n)/2+1

( p)

s ( p+1) (νs +n)/2+1

(1−z)

lp

z −(µ+1) (1−z)(νs =− lp

( p)

=

( p) f 0i (s ; z) ( p)

f 1i (s ; z)

, 0 < z < 1, s ∈ D,

Hν ( p+1) (z), s

( p+1) (−νs +n)/2

( p+1)

( p) f 01 (s ; z)

h ν ( p+1) (z),

= z −(µ+1) (1−z)

f 00 (s ; z) = −

( p) Fi (s ; z)

( p+1)

z(1−z)

+n)/2

+l +m −1 d νs + z dz 2 m(l + p + m) + (1−z) h ν ( p+1) (z), s l +m ( p+1)

z(1−z)

+l +m −1 z 2 lp − (1−z) Hν ( p+1) (z). s l +m

−νs d + dz

Then Fi (s ; z) (i = 0, 1) is a C ∞ -solution of (29) such that −m(l+ p+m)/(lp(l+m)) ( p) (33) F0 (s ; z) = 1 + O(s ; z) , 1 ( p+1) c(νs ) 1/(l+m) −(µ+1) ( p) F1 (s ; z) = z 1 + O(s ; z, z µ+1 log z) , µ+1 1


329

and such that for Re s > 0 and y = 1 − z, (34)

( p+1)

) −(νs( p+1) +l+m−1)/(2lp) (−νs( p+1) +n)/2 y 1 + O(s ; y) , = ( p+1) y νs ( p+1) −(−νs +l+m−1)/(2lp) (νs( p+1) +n)/2 ( p) F1 (s ; z) = y 1 + O(s ; y) , y c(νs

( p) F0 (s ; z)

( p)

Proof. By Proposition 3.3, we have only to check that the functions f 0i (s ; z) and ( p) f 1i (s ; z) are solutions of the equations (28) and (30) for each i = 0, 1. This is done by a direct computation. Using Lemma 3.7, we obtain (33) and (34). p ( p) Remark. The function F0 s 2 − 4l(m − 1); tanh 2 t is the A-radial part of an Eisenstein integral for H \G associated with the principal series IndGP0 (E p, p ; s), to be defined in the proof of Proposition 5.5. ( p)

( p)

By Propositions 3.3 and 3.8, F0 (s ; z) and F1 (s ; z) are two linearly independent solutions of (23). We proceed to construct two more solutions using Lemma 3.4. ( p) u (s ; z) ( p) Proposition 3.9. For i = 0, 1, define the function Ui (s ; z) = 0i , s ∈ D, ( p) u (s ; z) 1i 0 < z < 1, by ( p)

( p)

u 10 (s ; z) = (1 − z)(−νs ( p) u 11 (s ; z) ( p)

=z

u 00 (s ; z) =

−(µ+1)

+n)/2−1

s

( p)

(νs +n)/2−1

(1 − z)

Hν ( p) (z), s

( p) (−νs +n)/2−2

(1 − z)

lp

( p)

( p) u 01 (s ; z)

h ν ( p) (z),

z −(µ+1) (1 − z)(νs = lp

( p)

z(1 − z)

+n)/2−2

d νs + dz

− l − (m + 1) z 2 lp + (1 − z) h ν ( p) (z), s l +m ( p)

d −νs − l − (m + 1) + z dz 2 m(l + p + m) − (1 − z) Hν ( p) (z). s l +m

z(1 − z)

( p)

Then Ui (s ; z) is a C ∞ -solution of (31). ( p)

( p)

Proof. A direct computation shows u 0i (s z) and u 1i (s ; z) satisfy the two differential equations of Lemma 3.5. The conclusion then follows from Lemma 3.5. ( p)

Lemma 3.10. Let Fi (s ; z) be the solutions defined in Proposition 3.8 and form ( p) ( p) the matrix-valued function 8( p) (s ; z) = [F0 (s ; z) F1 (s ; z)] (0 < z < 1). Define

330

MASAO TSUZUKI

the function ( p) Ai (s ; z)

=

( p) a0i (s ; z) ( p)

a1i (s ; z)

,

˜ 0 < z < 1, s ∈ D,

for i = 0, 1, by ( p+1)

( p)

( p)

a00 (s ; z) = −z(1 − z)(νs −νs )/2−1 Hν ( p+1) (z)h ν ( p) (z) s s Z z ( p+1) ( p) + (1 − w)(νs −νs )/2−2 (1 + w) Hν ( p+1) (w)h ν ( p) (w) dw, s

0

( p+1)

( p)

s

( p)

a10 (s ; z) = z µ+2 (1 − z)(−νs −νs )/2−1 h ν ( p+1) (z)h ν ( p) (z) s s Z z ( p+1) ( p) − wµ+1 (1 − w)(−νs −νs )/2−2 (1 + w) h ν ( p+1) (w)h ν ( p) (w) dw, s

0

( p+1)

( p)

s

( p)

a01 (s ; z) = −z −µ (1 − z)(νs +νs )/2−1 Hν ( p+1) (z)Hν ( p) (z) s s Z z ( p+1) ( p) + w −(µ+1) (1 − w)(νs +νs )/2−2 (1 + w) Hν ( p+1) (w)Hν ( p) (w) dw, s

1

( p+1)

( p)

s

( p)

a11 (s ; z) = z(1 − z)(−νs +νs )/2−1 h ν ( p+1) (z)Hν ( p) (z) s s Z z ( p+1) ( p) − (1 − w)(−νs +νs )/2−2 (1 + w) h ν ( p+1) (w)Hν ( p) (w) dw. 0

s

s

These integrals are convergent, and for i = 0, 1, we have (35)

8( p) (s ; z)

d ( p) ( p) Ai (s ; z) = c(νs( p+1) )Ui (s ; z), dz

˜ 0 < z < 1, s ∈ D.

Proof. Lemma 3.7 shows that for a given compact set U in D there exists ε > 0 such that h s (z) and Hs (z) are bounded on (0, ε]∪[1−ε, 1) uniformly in s ∈ U . Hence the ( p) integrands of the formulas above except a01 (s ; z) are bounded on (0, ε] uniformly ( p) in s ∈ U , which implies the convergence of the integrals except a01 (s ; z). To obtain ( p) ˜ the convergence of a01 (s ; z) we need to assume s ∈ D. ( p) We prove (35). Since 1(z) = det 8 (s ; z) is the Wronskian for the funda( p) mental solutions Fi (s ; z) of the differential equation (29) it satisfies the relation (d/dz) 1(z) = tr(B( p) (z)) 1(z). Since −tr(B( p) (z)) = (n + 1) (1 − z)−1 + (µ + 1) z −1 , we easily obtain 1(z) = Cs z −(µ+1) (1 − z)n+1 with some constant Cs . Using (33), we have p+1 c(νs ) −(µ+1) 1(z) = − z 1 + O(s ; z, z log z) , lp

GREEN CURRENTS FOR MODULAR CYCLES ( p+1)

which implies that Cs = −c(νs 8( p) (s ; z)−1 = −

lp ( p+1) c(νs )

331

)/(lp). Hence

z µ+1 (1 − z)−(n+1)

"

( p)

( p)

f 11

− f 01

− f 10

f 00

( p)

#

( p)

(s ; z).

d ( p) ( p+1) ( p) A (s ; z) = c(νs ) 8( p) (s ; z)−1 Ui (s ; z). dz i ˜ set Proposition 3.11. For 0 < z < 1 and s ∈ D, Direct computation then gives

( p)

( p)

( p)

( p)

( p)

( p)

F˜ i (s ; z) = 8( p) (s ; z)Ai (s ; z) = F0 (s ; z) a0i (s ; z) + F1 (s ; z) a1i (s ; z). ( p) For each i = 0, 1, the function F˜ i (s ; z) is a C ∞ -solution of the differential equation (23) satisfying the equalities

( p+1) ) 1/(l+m) c(νs (36) = z 1+O(s ; z, z µ+1 log z) , µ+2 1 ( p) ( p+1) ) c(νs ) −m(l+ p+m)/(lp(l+m)) −µ c(ν ˜F( p) (s ; z) = − s z 1 µ(µ+1) 1 ( p) F˜ 0 (s ; z)

× 1+O(s ; z, z µ+1 log z, z 2µ+2 (log z)2 ) . When Re s > 0, setting y = 1 − z, we have ( p) a0 (s) (−νs( p) +n)/2 ) c(νs ) −2 y 1+O(s ; y) , (37) ( p) νs νs +l+m+1 1 ( p+1) −2c(νs ) a1 (s) (νs( p) +n)/2 ˜F( p) (s ; z) = y 1+O(s ; y) , 1 ( p) −νs +l+m+1 1 ( p) F˜ 0 (s ; z) =

( p+1)

c(νs

for holomorphic functions a0 (s), a1 (s) on C. ( p)

Proof. Lemma 3.4 combined with (35) shows that F˜ i (s ; z) is a solution of (23). The formulas (36) and (37) follow from (33), (34) combined with the asymptotics ( p) of ai j (s ; z) which is deduced from their explicit formula and Lemma 3.7. ( p) ( p) ( p) ( p) Proposition 3.12. The functions F0 (s ; z), F1 (s ; z), F˜ 0 (s ; z) and F˜ 1 (s ; z) constructed above form a fundamental system of solutions of the differential equa˜ tion (23) on 0 < z < 1, which depend holomorphically on s ∈ D.

Proof. The solutions of (23) form a 4 dimensional C-vector space. We already ( p) ( p) know that the functions Fi (s ; z) (i = 0, 1) and F˜ i (s ; z) (i = 0, 1) are solutions of (23) with linearly independent asymptotic behavior as z → +0.

332

MASAO TSUZUKI

The case p = 0. Proposition 3.13. For i = 0, 1, define F˜ i(0) (s ; z), 0 < z < 1, s ∈ D˜ by (1) νs −n νs(1) +λ , − + 1; µ + 1; z , F − 2 1 2 2 (1) (1) ν (1) +λ (1) ν −n (0) F˜ 1 (s ; z) = z −µ (1 − z)(νs +n)/2 2 F1 s + 1, s ; νs + 1; 1 − z . 2 2 (1)

(0)

F˜ 0 (s ; z) = (1 − z)(−νs

+n)/2

The F˜ i(0) (s ; z) form a fundamental system of C ∞ -solutions of (32) such that (0)

F˜ 0 (s ; z) = 1 + O(s ; z),

(38)

(0)

F˜ 1 (s ; z) =

c(νs(1) ) νs(1) +n −µ z 1 + O(s ; z, z µ log z) , µ(µ+1) 2

and such that for Re s > 0 and y = 1 − z c(νs(1) ) νs(1) +n (−νs(1) +n)/2 y 1 + O(s ; y) , (1) 2 (µ+1)νs (1) (0) F˜ 1 (s ; z) = y (νs +n)/2 1 + O(s ; y) . (0)

(39)

F˜ 0 (s ; z) =

˜ By Proof of Theorem 3.2. First we construct the family of functions φs (s ∈ D). the general theory, the map ψ : (h, t, k) 7→ hat k from H ×(0, ∞)×K onto G − H K is a submersion whose fibres are given as ψ −1 (hat k) = {(hm, t, m −1 k) | m ∈ M}. ( p) Hence we have a well-defined C ∞ -function φs : G − H K → F p, p for each 0 < p 6 µ by setting φs( p) (hat k) =

µ(µ + 1)lp ( p+1) ( p) c(νs ) c(νs )

( p) ( p) ( p) ( p) τ (k)−1 f˜01 (s ;tanh 2 t) v0 + f˜11 (s ;tanh 2 t) v1 .

( p) ( p) ( p) Here F˜ 1 (s ; z) is the column-matrix with entries f˜01 (s ; z) and f˜11 (s ; z), the solution of (23) constructed in Proposition 3.11. For p = 0 we also have a welldefined C ∞ -function φs(0) : G − H K → C such that

φs(0) (hat k) =

µ(µ + 1) c(νs(1) )

2 (0) F˜ 1 (s ; tanh 2 t), (1) νs + n

(h, t, k) ∈ H × (0, ∞) × K ,

where F˜ (0) is the function defined in Proposition 3.13. Define the function 1 (s ; z)V Pµ φs : G − H K → µ,µ p∗C by φs (g) := p=0 γ p L µ− p (φs( p) (g)) with {γ p } the family of real numbers in Proposition 1.4. Then, by Propositions 3.11 and 3.13, {φs }s∈ D˜ has the required properties. We prove the uniqueness of the family φs . Assume we are given another fam˜ possessing the same properties as φs . Then we can ily of functions ϕs (s ∈ D) Pµ ( p) ( p) µ− write ϕs (g) = p=0 γ p L p (ϕs (g)) with C ∞ -functions ϕs : G − H K → F p, p


333

uniquely along the decomposition (27). We shall show that for each 0 6 p 6 µ, ( p) ( p) ˜ by condition (i) of Theorem 3.2, φs (g) = ϕs (g) for all g ∈ G − H K and s ∈ D; we have only to show it assuming Re s > n. ( p) ( p) ( p) First consider the case p > 0. Then ϕs (at ) = g0 (tanh 2 t) v0 +g1 (tanh 2 t) v1 ( p) g (z) with a C ∞ -function G(z) = g01 (z) on 0 < z < 1. Since ϕs satisfies (22), the function G(z) is a solution of (23). Hence by Proposition 3.12, there are complex numbers c0 , c1 , d0 , d1 such that ( p)

( p)

( p)

( p)

G(z) = c0 F0 (s ; z) + c1 F1 (s ; z) + d0 F˜ 0 (s ; z) + d1 F˜ 1 (s ; z).

We examine the behavior as z → +0 on both sides of this identity to show c1 = 0. We have limz→+0 z µ+1 G(z) = 0 by condition (iv). By Propositions 3.8 and 3.11 we have ( p)

( p)

( p)

lim z µ+1 F0 (s ; z) = lim z µ+1 F˜ 0 (s ; z) = lim z µ+1 F˜ 1 (s ; z) = 0,

z→+0

z→+0

z→+0

( p)

lim z µ+1 F1 (s ; z) 6= 0.

z→+0

( p) ( p) ( p) Hence c1 = 0 and c0 F0 + d0 F˜ 0 + d1 F˜ 1 = G. Similarly we can use the behavior as y = 1 − z → +0 to conclude c0 = d0 = 0. Indeed, since we assume Re s > n, ( p) we have limt→+∞ e(−νs +n)t G(tanh 2 t) = 0 by condition (v), and ( p)

lim e(−νs

t→+∞

+n)t ( p) F0 (s ; tanh 2 t)

( p)

= lim e(−νs t→+∞

+n)t ˜ ( p) F1 (s ; tanh 2 t)

=0

( p) ( p) by Propositions 3.8 and 3.11. Hence d0 = 0 and c0 F0 + d1 F˜ 1 = G. Since Re νs( p) > Re νs( p+1) > Re s > n (see Lemma 3.1), we have ( p+1)

lim e(−νs

t→+∞

+n)t ˜ ( p) F1 (s ; tanh 2 t)

( p+1)

= lim e(−νs t→+∞

+n)t

G(tanh 2 t) = 0

by condition (v) and Proposition 3.11. Hence c0 = 0. ( p) Consequently, G = d1 F˜ 1 . By condition (iv) and Proposition 1.3, we have m(l + p + m)/(l + m) µ lim z G(z) = . −lp z→+0 At the same time, Proposition 3.11 gives ( p+1) ( p) c(νs )c(νs ) m(l+ p+m)/(l+m) = , −lp µ(µ+1)lp ( p+1) ( p) so the constant d1 equals µ(µ+1)lp/ c(νs )c(νs ) . ( p) ( p) ( p) From the definition of φs , we have ϕs (at ) = φs (at ). This is sufficient ( p) ( p) to conclude ϕs (g) = φs (g) for all g ∈ G − H K by the (H, K )-equivariance ( p) lim z µ F˜ 1 (s ; z) z→+0

334

MASAO TSUZUKI

condition. The discussion for the case p = 0 is quite similar. This completes the proof of uniqueness. Some properties of the secondary spherical function. ˜ be the secondary spherical function constructed in Theorem 3.14. Let φs (s ∈ D) Theorem 3.2. (a) There exist µ polynomial functions aα (s) (0 6 α 6 µ − 1) with values in Vµ,µ ∗ M Vµ,µ ∗ M pC , positive number ε and pC -valued holomorphic func˜ tions bi(s,z) (i = 0, 1, 2) on {(s, z) | s ∈ D, |z| < ε} such that a0 (s) = w,

aα (−s) = aα (s),

deg(aα (s)) 6 2α,

˜ z = tanh 2 t ∈ (0, ε), and such that for s ∈ D, (40)

φs (at ) =

µ−1 X α=0

aα (s)

z µ−α

+ b0 (s ; z) + b1 (s ; z) log z + b2 (s ; z) z µ+2 (log z)2 .

Vµ,µ ∗ M (b) There exists a positive number ε0 and pC -valued holomorphic func( p) tions f (s ; y) (0 6 p 6 µ) on {(s, y) | |y| < ε 0 , Re s > n} such that

(41)

φs (at ) =

µ X

( p)

y (νs

+n)/2 ( p)

f

(s ; y),

Re s > n, y =

p=0

1 cosh 2 t

∈ (0, ε0 ).

˜ the func(c) For any differential operator with holomorphic coefficient ∂s on D, ∞ tion ∂s φs (g) on G − H K belongs to Cτ . Proof. (a) Recall the construction of φs . Then (36) and (38) immediately yields the existence of the expression of the form (40) except the nature of the functions aα (s). We have a0 (s) = w from condition (iv). Hence it remains to show that aα (s) is an even polynomial function with degree no more than 2α. For that purpose, we examine the differential equation (22) again. By Theorem 2.3, equation (22) can be written as 2 d µ+1 n−1 d (42) + + + Q(s ; z) φs = 0, dz 2 z 1 − z dz with Q(s ; z) =

S0 τ ( M0 ) + λ2 − s 2 −S + 2 + . z(1 − z) z (1 − z) 4z(1 − z)2

We have Q(s ; z) =

S0

z2

+

∞ X γ =−1

Qγ (s) z γ ,

|z| < 1,


335

with Qγ (s) = −S + S0 + 14 (γ + 2) λ2 − s 2 + τ ( M0 ) ,

γ > −1.

Substitute this expression of Q(s ; z) and (40) to the left-hand side of (42) and compute the coefficient of z −µ+α−2 . Since it should be zero, we obtain the recurrence relation among the tensors aα (s): for 0 < α 6 µ − 1, (43)

S − α(µ − α) aα (s) = − 0

α−1 X

Qα−γ −2 (s) − (n − 1)(µ − γ ) aγ (s).

γ =0 M ) (0 < p 6 µ) and the By (17), the operator S0 preserves the subspaces L µ− p (F p, p ( p) restriction of S0 is represented with respect to the basis L µ− p (vi ) (i = 0, 1) by   pl m(l + p + m) − −  l +m (l + m)2    ( p) Q−2 =  .  m(l + p + m)  −lp − l +m ( p) Since det Q−2 −α(µ−α) = α(α+1)(α−µ−1)(α−µ) is not zero for 0 < α 6 µ−1, V the operator S0 −α(µ−α) is invertible on µ,µ p∗C M . Therefore we can determine aα (s) uniquely by the recurrence relation (43), from which we easily know that aα (s) is an even polynomial function with degree no more than 2α. By the construction of φs , part (c) is obvious. Part (b) follows from (37) and (39).

˜ define the function ψs : G − H K → The function ψs . For s ∈ D, ψs (g) = ∇− ∇+ φs (g),

Vr,r

p∗C by

g ∈ G − H K.

We easily see that ψs has the same properties as (2) and (3) of the previous theorem for φs . It also behaves similarly to (40) for φs near t = 0, (44)

ψs (at ) =

µ−1 X α=0

cα (s)

z µ−α

+ O(s; 1, log z, z µ (log z)2 ),

except that the degree of the ( r,r p∗C ) M -valued polynomial cα (s) is no more than 2(α + 1). Indeed by applying the expression (24) of the Schmid operator to (40) we can obtain the required expression with one extra term of the form uz −(µ+1) . The only thing we have to do here is to show u = 0. By a direct computation, we have u = (µ(µ + 1)e(ω0 ∧ ω0 ) + µA + R− R+ )w. Since w is (H ∩ K )-invariant and the elements Z j , Z j with √ n −r +1 6 √ j 6 n − 1 belong to (k ∩ h)C , we have R± w = 0. Hence u = µ(2/ −1) η − ( −1/2) µ ω0 ∧ ω0 ∧ w. By Proposition 1.4, the right-hand side of this identity is zero. V

336

MASAO TSUZUKI

4. Poincaré series Let 0 be a discrete subgroup of G. We assume that (G, H, 0) is arranged as follows. There exists a connected reductive Q-group G, a closed Q-subgroup H of G and an arithmetic subgroup 1 of G(Q) such that there exists a morphism of Lie groups from G(R) onto G with compact kernel which maps H(R) onto H and 1 onto 0. Set K H = H ∩ K , 0 H = 0 ∩ H . Invariant measures. Let dk and dk0 be the Haar measures of compact groups K and K H with total volume 1. There is a unique Haar measure dg on G such that the quotient measure dg/dk corresponds to the measure on the symmetric space G/K determined by the invariant volume form vol. We define dh on H analogously: dh/dk0 corresponds to the measure on H/K H determined by vol H . Lemma 4.1. For any measurable functions f on G we have Z Z Z Z ∞ f (g) dg = dh dk f (hat k)%(t) dt G

H

K

0

with dt the usual Lebesgue measure on R and %(t) = 2cr (sinh t)2r −1 (cosh t)2n−2r +1 ,

cr = π r /µ!.

Proof. For closed subgroups Q 1 ⊂ Q 2 of G with Lie algebras qi = Lie(Q i ) for i = 1, 2, we regard (q2 /q1 )∗ ⊂ g∗ by the dual map of the composite of the orthogonal projection g → q2 and the canonical surjection q2 → q2 /q1 . Let volq2 /q1 be the V element ξ1 ∧ · · · ∧ ξs ∈ (q2 /q1 )∗ with {ξi } any orthonormal basis of (q2 /q1 )∗ . Assume Q 1 is compact. Then there exists a unique left Q 2 -invariant s-form Z Q 2 /Q 1 on Q 2 /Q 1 whose value at o = eQ 2 is volq2 /q1 . Let d Z Q 2 /Q 1 be the Q 2 -invariant measure on Q 2 /Q 1 corresponding to Z Q 2 /Q 1 . For example, volg/k = vol and volh/m = vol H ∧ volh∩k/m . The decomposition G = HAK yields the diffeomorphism j from H/M ×(0, ∞) ˙ t) = hat K (h˙ ∈ H/M, t > 0); see [Heckman to (G − H K )/K defined by j (h, and Schlichtkrull 1994, Theorem 2.4, p. 108]. A simple computation with the aid of the formulas in the proof of Lemma 2.2 proves the identity j ∗ ZG/K = 2(sinh t)2r −1 (cosh t)2n−2r +1 Z H/M ∧ dt. Hence under the identification H/M × (0, ∞) = (G − H K )/K we have (45)

˙ dt. d Z G/K (g) ˙ = 2(sinh t)2r −1 (cosh t)2n−sr +1 d Z H/M (h)

The measure d Z G/K (g) ˙ is precisely dg/dk in our normalization. Let dm be the Haar measure of M with total volume one. The resulting quotient measure dh/dm


337

˙ We determine the proportionality on H/M should be proportional to d Z H/M (h). constant C0 in such a way that (46)

dh ˙ = C0 d Z H/M (h). dm

We have a decomposition ˙ = dh d Z H ∩K /M (k˙0 ), d Z H/M (h) dk0 ˙ = dh/dk0 . Using the relation dh/dm = (dh/dk0 ) (dk0 /dm) , since d Z H/H ∩K (h) it follows that dk0 = C0 d Z H ∩K /M (k˙0 ). dm Since the total measure R of H ∩ K /M with respect to dk0 /dm is 1, we then obtain −1 the equality C0 = H ∩K /M d Z H ∩K /M (k˙0 ). To compute this integral, we use the diffeomorphism 5 : H ∩ K /M → S2r −1 defined by 5(diag(u 1 , u 2 , u 0 ) M) = u −1 (u 1 , u 2 , u 0 ) ∈ U(n − r ) × U(r ) × U(1), 0 u 2 (e), P with S2r −1 := z ∈ Cr ri=1 |z i |2 = 1 and e := t (0, . . . , 0, 1) ∈ S2r −1 . Thus, easily, P2r the pullback of the volume form ZS2r −1 = i=1 (−1)i−1 xi d x1 ∧· · ·∧ dˆ xi ∧. . . d x2r of the sphere S2r −1 is 2Z H ∩K /M . Hence Z Z πr 1 −1 ˙ d Z S2r −1 = . C0 = d Z H ∩K /M (k0 ) = 2 S2r −1 0(r ) H ∩K /M Using (45), (46) and the value of C0 just obtained, the required integration formula easily follows. Currents defined by Poincaré series. Let F denote the set of the families of func˜ or ϕs = ∂s ψs (s ∈ D) ˜ with some differential tions {ϕs }s∈ D˜ such that ϕs = ∂s φs (s ∈ D) ˜ operator ∂s with holomorphic coefficient on D. For {ϕs } ∈ F, introduce the Poincaré series X ˜ s )(g) = (47) P(ϕ ϕs (γ g), g ∈ G, γ ∈0 H \0

which is the most basic object in our investigation. First of all, we discuss its convergence in a weak sense. Note that ϕs takes its values in a finite dimensional V Hilbert space p∗C with the norm kαk = (α|α)1/2 . Theorem 4.2. The function in s defined by the integral X Z ˜ P(kϕ kϕs (γ g)k d g˙ s k) := 0\G

γ ∈0 H \0

338

MASAO TSUZUKI

is locally bounded on Re s > n. For each s with Re s > n, the series (47) converges absolutely almost everywhere in g ∈ G to define an L 1 -function on 0\G. Proof. We assume ϕs = ∂φs . (The proof for ∂ψs is the same.) Let U be a compact subset of Re s > n. Then by Theorem 3.14 and Lemma 3.1, there exist positive numbers a, δ and C0 such that kϕs (at )k 6 C0 t −2µ ,

(s, t) ∈ U × (0, a],

kϕs (at )k 6 C0 e−t (2n+δ) ,

(s, t) ∈ U × [a, ∞).

From the form of %(t), we can find a positive constant C1 such that %(t) 6 C1 t 2µ+1 for t ∈ (0, a]

and%(t) 6 C1 e2nt for

t ∈ [a, ∞).

First as in the proof of [Oda and Tsuzuki 2003, Proposition 3.1.1] and then by using ˜ the estimations above, we obtain the uniform bound of P(kϕ s k) for s ∈ U . If 0 is neat, then the quotient space 0\G/K acquires a structure of complex manifold from the one on G/K ∼ = D. Let π : G/K → 0\G/K be the natural projection. Let A(0\G/K ) denote the space of C ∞ -differential forms on 0\G/K and Ac (0\G/K ) the subspace of compactly supported forms. Given α ∈ A(0\G/K ), V there is a unique C ∞ -function α˜ : G → p∗C such that α(γ ˜ gk) = τ (k)−1 α(g) ˜ for γ ∈ 0 and k ∈ K and such that V V (48) h(π ∗ α)(gK ), d L g (ξo )i = hα(g), ˜ ξo i, g ∈ G, ξo ∈ p = To (G/K ). Here L g denotes the left translation on G/K by the element g and we identify p with To (G/K ), the tangent space of G/K at o = eK . Then for any α, β ∈ A(0\G/K ), we have Z Z ˜ (49) α ∧ ∗β = (α(g)| ˜ β(g)) d g. ˙ 0\G/K

0\G

For any left 0-invariant function f on G, the integral Z J H ( f ; g) = f (hg) dh, g ∈ G, 0 H \H

plays a fundamental role in our further study. We already discussed the convergence problem of this integral in [Oda and Tsuzuki 2003, §3.2]. For convenience we recall the result. If 0 is cocompact, we take a compact fundamental domain S1 for 0 in G and tS1 the constant function 1. Hence G = 0S1 in this case. If 0 is not cocompact, then one can fix a complete set of representatives Pi (1 6 i 6 h) of 1conjugacy classes of Q-parabolic subgroups in G together with Q-split tori Gm ∼ = Ai i in the radical of P such that an eigencharacter of Ad(t) (t ∈ Gm ) in the Lie algebra of Pi is one of t j ( j = 0, 1, 2). For each i, let T i and N i be the images in G of Ai (R) and the unipotent radical of Pi (R) respectively. Then we can choose a Siegel


339

domain Si in G with respect to the Iwasawa decomposition G = N i T i K for each i such that G is a union of 0Si (1 6 i 6 h). Let tSi : Si → (0, ∞) be the function tSi (n i t i k) = t (n i t i k ∈ Si ). Here t i denotes the image of t ∈ Gm (R) ∼ = Ai (R) in Ti. Given δ ∈ (2r n −1 , 1), let Mδ be the space of all left 0-invariant C ∞ -functions V f : G → p∗C with the K -equivariance f (gk) = τ (k)−1 f (g) such that, for any ε ∈ (0, δ) and D ∈ U (gC ), we have kR D ϕ(g)k ≺ tSi (g)(2−ε)n ,

g ∈ Si , 1 6 i 6 h.

Proposition 4.3. Let f ∈ Mδ with δ ∈ (2r n −1 , 1) and D ∈ U (gC ). (1) For any ε ∈ (2r n −1 , δ) J H (kR D f k; at ) ≺ e(2−ε)nt ,

t > 0.

The function J H ( f ; g) is of class C ∞ , belongs to Cτ∞ and J H (R D f ; g) = R D J H ( f ; g),

g ∈ G.

(2) For any {ϕs } ∈ F, the integral Z ˜ s )(g)|R D f (g))| d g˙ |( P(ϕ 0\G

is finite if Re s > 3n − 2r , and Z Z ˜ ( P(ϕs )(g)|R D f (g)) d g˙ = 0\G

∞

%(t) (ϕs (at )|R D J H ( f ; at )) dt. 0

Proof. See [Oda and Tsuzuki 2003, Theorem 3.2.1] and its proof. The number τQ (G, σ ) is 2(1 − r n −1 ) in our setting here. Proposition 4.4. There exists a unique current P(ϕs ) on 0\G/K such that for α ∈ Ac (0\G/K ), Z Z ∞ ˜ (50) hP(ϕs ), ∗αi = ( P(ϕs )(g)|α(g))d ˜ g˙ = %(t) (ϕs (at )|J H (α˜ ; at )) dt. 0\G

0

˜ Then for any α ∈ Ac (0\G/K ), Let ∂s be a holomorphic differential operator on D. ˜ the function s 7→ hP(ϕs ), αi is holomorphic on D and ∂s hP(ϕs ), αi = hP(∂s ϕs ), αi. ˜ s ) on 0\G satisfies P(ϕ ˜ s )(gk) = τ (k)−1 P(ϕ ˜ s )(g) Proof. The L 1 -function P(ϕ 1 for all k ∈ K . Hence it defines an L -form P(ϕs ) on 0\G/K naturally. Then R the linear functional α 7→ 0\G/K P(ϕs ) ∧ α on Ac (0\G/K ) is a current, which we also denote by P(ϕs ). The first equality of (50) follows from (49). Let α ∈ Ac (0\G/K ). Since α˜ is left 0-invariant and has compact support mod 0, the V image α(0\G) ˜ is a compact subset of p∗C . Hence there is a positive constant C0

340

MASAO TSUZUKI

such that |(u|α(g))| ˜ 6 C0 kuk for all u ∈ p∗C and g ∈ G. Then for any loop C in Re s > n, we have Z Z Z Z X X ds |(ϕs (γ g)|α(γ ˜ g))| d g˙ 6 C0 ds kϕs (γ g)k d g˙ V

C

0\G γ ∈0 \0 H

C

Z = C0

0\G

γ ∈0 H \0

˜ P(kϕ s k) ds.

C

By Theorem 4.2, the last integral in s is finite. Hence by Fubini’s theorem and by the holomorphicity of each function s 7→ (ϕs (g)|α(g)), ˜ we have Z Z ˜ s )(g)|α(g)) ds ( P(ϕ ˜ d g˙ = 0. 0\G

C

Since C is arbitrary, the holomorphicity in s of the integral (50) follows. The second equality of (50) follows by a standard computation. Differential equation of functions related to G˜ s . For each 0 6 j 6 µ, put δ j,s G˜ s = ˜ s, j φs ) with (cr µ µ!)−1 P(δ 1 1 d j δ j,s = , j ∈ N, j! 2s ds which we will regard as a

Vµ,µ

p∗C -valued L 1 -function on 0\G.

Theorem 4.5. Let 0 6 j 6 µ be an integer. Suppose Re s > 3n − 2r . Then for any f ∈ Mδ with δ ∈ (2r n −1 , 1), we have Z 4 (δ j,s G˜ s (g)|R(+λ2 −¯s 2 ) j+1 f (g)) d g˙ = (w|J H ( f ; e)). µ! 0\G Proof. For f ∈ Mδ , let I j ( f ) denote the integral above. Then we shall prove 4 I j ( f ) = I j−1 ( f ) for 1 6 j 6 µ and I0 ( f ) = µ! (w|J H ( f ; e)), which obviously imply the equality. Since we already have Proposition 4.3, we can set aside the convergence argument of various integrals in these formulas. Fix 0 6 j 6 µ and put ˜ F(g) = J H (R(−s 2 +λ2 ) j f ; g), g ∈ G. Since F˜ ∈ Cτ∞ , (14) implies ˜ t ) = (D + λ2 − s¯ 2 )F(t) R(+λ2 −¯s 2 ) F(a with D the differential operator in t > 0 given by the formula inside the bracket of ˜ t ). By Proposition 4.4(2), we have (14), and F(t) = F(a Z ∞ 1 (51) Ij( f ) = %(t) δ j,s φs (at )|(D + λ2 − s¯ 2 )F(t) dt. cr µ µ! 0


341

Let 0 < ε < R. Then performing integration-by-parts and using the fact that the operators S, S0 and τ ( M0 ) are self-adjoint, we obtain R

Z (52)

ε

%(t) δ j,s φs (at )|(D + λ2 − s¯ 2 )F(t) dt Z R = −8(R) + 8(ε) + %(t) (D + λ2 − s 2 )δ j,s φs (at )|F(t) dt ε

with 8(t) = %(t) δ j,s φs (at )|F 0 (t) − %0 (t) δ j,s φs (at )|F(t) d δ j,s φs (at )|F(t) − %(t) dt + %(t) (2n − 2r + 1) tanh t + (2r − 1) coth t δ j,s φs (at )|F(t) . Now we use (40). Since the function aα (s) appearing there is an even polynomial function of degree no more than 2α, we have δ j,s φs (at ) =

wµ− j

z µ− j

(1 + O(s ; z, z log z))

with wα = (δ α,s aα )(0). Using this, we easily see that 8(t) = 4cr (µ − j) wµ− j |F(0) z j 1 + O(s ; z, z log z) to conclude that (53)

lim 8(ε) = 4cr µδ0 j (w|F(0))

ε→+0

with δ0 j the Kronecker delta. To compute the limit of 8(R) as R goes to infinity, we use (41) and Lemma 3.1. After a computation we obtain (54)

lim 8(R) = 0.

R→+∞

Here we need the inequality Re s > n which follows from Re s > 3n − 2r . A simple computation (or [Gon and Tsuzuki 2002, Claim 3.1.6, p. 240]) shows the first equality of (55)

(D + λ2 − s 2 )δ j,s φs (at ) = δ j,s (D + λ2 − s 2 )φs (at ) + δ j−1,s φs (at ) = δ j−1,s φs (at ),

and the differential equation of (22) gives the second equality. Putting together equations (51)–(55), we obtain the result.

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5. Spectral expansion In this section we investigate the spectral expansion of the functions δ j,s G˜ s to obtain a meromorphic continuation of the current-valued function s 7→ G s , which is already holomorphic on the half plane Re s > n. Spectral expansion. In order to describe the spectral decomposition of the function δ µ,s G˜ s , we need some preparations. q For positive q, let L0 (τ ) denote the Banach space of all measurable functions V ∗ K , f (γ gk) = τ (k)−1 f (g) and Rf : G → qpC such that for all γ ∈ 0 and qk ∈ (d) ˙ < ∞. For 0 6 d 6 n, let L0 (τ ) denote the subspace of those 0\G k f (g)k d g V q functions f ∈ L0 (τ ) with values in d,d p∗C . The inner product of two functions f 1 and f 2 in L20 (τ ) is given as Z h f1| f2i = ( f 1 (g)| f 2 (g)) d g. ˙ 0\G

For each 0 6 d 6 n, let {λ(d) n }n∈N be the increasing sequence of the eigenvalues of the bidegree (d, d)-part of the Laplacian e 4 = −R such that each eigenvalue occurs with its multiplicity. Choose an orthonormal system {α˜ n(d) }n∈N in L20 (τ )(d) ˜ n(d) for each n. From now on we assume 0 is not cocompact. such that e 4α˜ n(d) = λ(d) n α Recall the parabolic subgroups P i used to construct the Siegel domains Si (page 338). Let P i = M0i T i N i be its Langlands decomposition with M0i := Z K (T i ). For each i let 0 P i = 0 ∩ P i and 0 M i = M0i ∩ (0 P i N i ). Then 0 M i is a finite group 0 0 which is trivial if 0 is torsion free. Vd,d ∗ 0 M i pC 0 and a complex number s, define the For a vector u ∈ Vi(d) := function ϕsi (u ; g) on G using the Iwasawa decomposition G = N i T i K by ϕsi (u ; n i t i k) = t s+n τ (k)−1 u, for n i ∈ N i , t > 0, k ∈ K . The Eisenstein series associated with u is defined by the absolutely convergent infinite series X (56) Ei (s ; u ; g) = ϕsi (u ; γ g), g ∈ G, γ ∈0 P i \0

if Re s > n. There exists a family of linear maps Ei (s) from Vi(d) to the space of automorphic forms on 0\G, which depends meromorphically on s ∈ C and is holomorphic on the imaginary axis, such that (Ei (s)(u))(g) = Ei (s ; u ; g) coincides with (56) when Re s > n. If u ∈ Vi(d) is an eigenvector of the Casimir operator M i 0 of M0i with eigenvalue c ∈ C, then R E(s; u) = (s 2 − n 2 + c) Ei (s; u) for any s ∈ C where Ei (s) is regular. It is also known that there exists a meromorphic family of operators {cij (s)} with cij (s) ∈ HomC Vi(d) , V(d) satisfying the functional equations j


E (−s) =

h X

E j (s) ◦ cij (s),

1 6 i 6 h,

cij (−¯s )∗ ◦ cij (s) = IdV(d) ,

1 6 i 6 h.

i

343

j=1

(57)

h X

i

j=1

Lemma 5.1. For 0 6 p 6 d and ε ∈ {0, 1}, let Wi(d) ( p ; ε) be the eigenspace of τ ( M i ) on Vi(d) corresponding to the eigenvalue (2 p − ε)(2n − 2 p + ε). Then we 0 have the orthogonal decomposition (d)

Vi

=

d M M

(d)

Wi ( p ; ε).

p=0 ε∈{0,1}

Proof. First we recall the construction of invariant tensors given in Proposition 1.2. For each p, q, let E p,q be the M0 -module defined in the proof of that proposition and T p,q be the M0 -homomorphism constructed there. For convenience we set E p,q = {0} when p or q is negative. Choose a parabolic subgroup P0 of G with Levi subgroup M0 A. Then we can find an element k˜i ∈ K such that P i = k˜i P0 k˜i−1 , M0i = k˜i M0 k˜i−1 , and T i = k˜i Ak˜i−1 . For p ∈ N, put p(+) := p and p(−) := p − 1. For ε, ε0 ∈ {+, −}, set E p,εε0 = T p, p (E p(ε), p(ε0 ) ); then E ip,ε,ε0 := τ (k˜i )(E p,εε0 ) is an irreducible sub M0i -module of F p, p . For each 0 6 p < 2n, the space F p, p , when considered as an M0i -module, is decomposed to the orthogonal direct sum of four subspaces E ip,εε0 with ε, ε0 ∈ {+, −}. The operator τ ( M i ) acts on E ip,εε0 by the 0 scalar ( p(ε) + p(ε0 ))(2n − p(ε) − p(ε0 )). Now put (58)

(d)

0Mi

(d)

0Mi

Wi ( p ; 0) = L d− p (E ip,++ )

Wi ( p ; 1) = L d− p (E ip,+− )

0

0

0Mi

⊕ L d− p−1 (E ip+1,−− ) 0Mi

⊕ L d− p (E ip,−+ )

0

0

,

.

Then the operator τ ( M i ) acts on Wi(d) ( p ; ε) for 0 6 p 6 d and ε ∈ {0, 1} as the 0 scalar (2 p − ε)(2n − 2 p + ε). By the K -decomposition (27), the space Vi(d) is decomposed as the orthogonal direct sum of those subspaces Wi(d) ( p ; ε). Lemma 5.2. (a) The eigenfunctions α˜ m(d) lie in the space Mδ for any δ ∈ (2r n −1 ,1). For each 0 6 j 6 µ, we have Z (µ) 4 (w|J H (α˜ m ; e)) (59) . (δ j,s G˜ s (g)|α˜ m(µ) (g)) d g˙ = (µ) 0\G µ! (λ2 − λm − s 2 ) j+1 (b) Let U be a compact subset on which the function Ei (s) is holomorphic. Then for all u ∈ Vi(d) , the image Ei (s)(u) is in Mδ for any δ ∈ (2r n −1 , 1) such that (µ) sups∈U |Re s| < 1 − δ. If u ∈ Vi satisfies τ ( M i )u = c u, we have, for each 0

344

MASAO TSUZUKI

0 6 j 6 µ, Z (60) (δ j,s G˜ s (g)|Ei (ν ; u, ; g)) d g˙ = 0\G

4 (w|J H (Ei (ν ; u); e)) µ! (λ2 − n 2 + c − s 2 + ν 2 ) j+1

for s ∈ C with |Re s| < 1 − 2r n −1 . Proof. The estimation in [Oda and Tsuzuki 2003, Lemma 3.3.1], which implies the first assertion of (a), is valid for our α˜ m(d) without modification. The first assertion of (b) follows from [Miatello and Wallach 1989, Lemma A.2.2]. The argument to prove (59) and (60) by Theorem 4.5 is the same as that in the proof of [Oda and Tsuzuki 2003, Proposition 6.2.2]. For each index (d, i, p, ε), fix an orthonormal basis Bi(d) ( p ; ε) of the space (d) Wi ( p ; ε). Theorem 5.3. Let Re s > 3n − 2r . Then there exists ε > 0 such that the function (µ) ˜ δ µ,s G˜ s (g) belongs to the space L2+ε 0 (τ ) . The spectral expansion of δ µ,s G s is δ µ,s G˜ s = Gdis (s) +

µ X X

G(c p,ε) (s)

p=0 ε∈{0,1}

with (61)

Gdis (s) = G(c p,ε) (s) =

Z ×

(µ) ∞ X 4 w|J H (α˜ m ; e) (µ) 2 2 r m=0 µ! (λ − λm − s )

1 √ 4π −1 h X X

√ −1 R i=1 (µ) u∈Bi ( p ;ε)

α˜ m(µ) ,

4 w|J H (Ei (ζ ; u); e) Ei (ζ ; u) dζ, µ! (λ2 − (n−2 p+ε)2 − s 2 + ζ 2 )r

where the summations in the right-hand side of this formula are convergent in L20 (τ )(µ) . Proof. Since the differential operator δ µ,s annihilates the even polynomial functions a j (s) (0 6 j 6 µ − 1) appearing in (40), the behavior of δ µ,s φs (at ) near t = 0 is given as δ µ,s φs (at ) = O(s ; 1, log z, z µ+2 (log z)2 ). By (41), the function δ µ,s φs (at ) has an exponential decay as t → +∞ when Re s > n. Hence we can argue in exactly the same way as [Oda and Tsuzuki 2003, §§5.2, 5.3] to obtain (µ) with some ε > 0. (In this reference, the results the estimate δ µ,s G˜ s ∈ L2+ε 0 (τ ) in Section 4, on which the arguments in §§5.2 and 5.3 rely, remain valid for our (G, H, 0) without modification.) Once we establish the L 2+ε -estimate of δ µ,s G˜ s , we can work out the spectral expansion of δ µ,s G˜ s using (59) and (60) by the same argument as in [Oda and Tsuzuki 2003, 6.2].


345

Some properties of Eisenstein period. Proposition 5.4. For 1 6 i 6 h and u ∈ Vi(d) , there exists a unique d,d p∗C -valued meromorphic function PiH (s ; u) on C which is regular and has the value given by the absolutely convergent integral J H (Ei (s ; u); e) at any regular point s ∈ C of Ei (s ; u) in |Re s| < 1 − 2r n −1 . V

Proof. This can be proved by the same argument as in [Oda and Tsuzuki 2003, Proposition 6.1.1]. Proposition 5.5. Let 1 6 i 6 h and 1 6 p 6 d. Then for any u ∈ Wi(d) ( p ; 1), we have PiH (s ; u) = 0 identically. Proof. We freely use the notation introduced in the proof of Lemma 5.1. For any unitary representation (E, σ ) of M0 and s ∈ C, let IndGP0 (E ; s) be the Fréchet space of all C ∞ -functions f : G → E satisfying the relation f (mat ng) = e(n+s)t σ (m) f (g) for any m ∈ M0 , at ∈ A and n ∈ N . The group G acts by the right translation on the space IndGP0 (E ; s) smoothly. For ε, ε0 ∈ {+, −}, 1 6 i 6 h, 0 6 p 6 d, put Uip,εε0 := Hom0 M i (E ip,εε0 , C). 0

Then by the theory of Eisenstein series there exists a meromorphic family of continuous G-homomorphisms i Eip,εε0 (s) : IndG P0 (E p,εε 0 ; s) ⊗ U p,εε0 → A(0\G),

which is given by the absolutely convergent sum X (Eip,εε0 (s)( f ⊗ u))(g) ˇ = u(τ ˇ (k˜i ) f (k˜i−1 γ g)),

s ∈ C,

g ∈ G,

γ ∈0 P i \0

for f ∈ IndGP0 (E p,εε0 ; s) and uˇ ∈ Uip,εε0 when Re s > n. Here A(0\G) denotes the space of (not necessarily K -finite) automorphic forms on G. By slightly extending the argument in the proof of [Oda and Tsuzuki 2003, Proposition 6.1.1], we can show that the integral 8ip,εε0 (s)( f ⊗ u) ˇ := J H (Eip,εε0 (s)( f ⊗ u); ˇ e) is convergent in some neighborhood of the imaginary axis and the function 8ip,εε0 (s) has a meromorphic continuation to C. The linear map 8ip,εε0 (s) : IndGP0 (E p,εε0 ; s) ⊗ Uip,εε0 → C is continuous and H -invariant, that is, 8ip,εε0 (s) is an H -spherical distribution. The maps Eip,εε0 (s) and 8ip,εε0 (s) naturally induce the linear maps Vd,d V Eip,εε0 (s)∗ : Hom K pC , IndGP0 (E p,εε0 ;s) ⊗ Uip,εε0 → Hom K d,dpC ,A(0\G) , V V 8ip,εε0 (s)∗ : Hom K d,d pC , IndGP0 (E p,εε0 ;s) ⊗ Uip,εε0 → d,d p∗C .

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MASAO TSUZUKI

On the other hand, we define the linear map Vd,d i J p,εε pC , IndGP0 (E p,εε0 ; s) ⊗ Uip,εε0 → Vi(d) 0 : Hom K by composing the maps Hom K

Vd,d

pC , IndGP0 (E p,εε0 ; s) ⊗ Uip,εε0

J1

Vd,d

pC , Ind KM0 (E p,εε0 ⊗ Uip,εε0 ) J3 Vd,d V pC , Ind KM i (E ip,εε0 ⊗ Uip,εε0 ) → Hom M i d,d pC , E p,εε0 ⊗ Uip,εε0

→ Hom K J2

→ Hom K

0

0

J4

→ Hom M i

0

Vd,d

M0i

pC , Ind0

J5 V (d) (10 M i ) → Hom0 M i d,d pC , 10 M i ∼ = Vi , i

M0

0

0

0

where J1 is the isomorphism induced by the natural identification of the K -modules IndGP0 (E p,εε0 ; s) ⊗ Uip,εε0 ∼ = Ind KM0 (E p,εε0 ⊗ Uip,εε0 ), the map J2 is induced by the K -isomorphism Ind KM0 (E p,εε0 ) ∼ = Ind KM i (E ip,εε0 ) that assigns the function 0

k 7→

τ (k˜i ) f (k˜i−1 k)

∈

Ind KM i (E ip,εε0 ) 0

to

f (k) ∈ Ind KM0 (E p,εε0 ),

J3 and J5 are the isomorphisms giving the Frobenius reciprocity, and J4 is the inclusion induced by the map E ip,εε0 ⊗ Uip,εε0 ,→ C ∞ (0 M0i \M0i ) which identifies the tensor u ⊗ uˇ ∈ E ip,εε0 ⊗ Uip,εε0 with the function m 7→ hu, ˇ τ (m)ui on 0 M0i \M0i . i The map J p,εε 0 is injective and has the defining formula i hJ p,εε ˇ ξ i = hu, ˇ τ (k˜i )(α(ξ )(k˜i−1 ))i 0 (α ⊗ u), V V for ξ ∈ d,d pC , α ∈ Hom K d,d pC , IndGP0 (E p,εε0 ; s) , and uˇ ∈ Uip,εε0 . From

(d) i i i ( p ; 1) = Im J p,+− ⊕ Im J p,−+ . The the definition of J p,εε 0 and (58), we get Wi formula i i Ei (s ; u) = (Eip,+− (s)∗ ⊕ Eip,−+ (s)∗ ) ◦ (J p,+− ⊕ J p,−+ )−1 (u),

(d)

u ∈ Wi ( p ; 1),

which follows directly from the definitions when Re s > n, remains valid as an identity of meromorphic functions on C. When |Re s| is sufficiently small, the integration on 0 H \H of this formula yields yet another formula: for u ∈ Wi(d) ( p ; 1), (62)

i i J H (Ei (s ; u); e) = (8ip,+− (s)∗ ⊕ 8ip,−+ (s)∗ ) ◦ (J p,+− ⊕ J p,−+ )−1 (u).

By the general theory, there exists an open dense set U such that for s ∈ U the space M0 ∩H M C −∞ IndGP0 (E p,q ; s) H of H -spherical distributions is isomorphic to E p,q = E p,q [Heckman and Schlichtkrull 1994, Theorem 6.4, p. 151], which is zero when p 6= q by the Claim in the proof of Proposition 1.2. Hence 8ip,+− (s) = 0 and 8ip,−+ (s) = 0 for s ∈ U . This, combined with (62), implies J H (Ei (s ; u); e) = 0 for generic s. By analytic continuation we obtain PiH (s ; u) = 0 identically.


347

Lemma 5.6. For each 0 6 p 6 µ, put h

4 X (63) E˜ (µ) p (ν ; g) = µ! i=1

X

(w|PiH (−¯ν ; u)) Ei (ν ; u ; g),

g ∈ G, ν ∈ C.

(µ) u∈Bi ( p ;0)

(µ)

(µ)

Then E˜ p (ν ; g) is independent of the choice of the orthonormal basis Bi ( p ; 0) (µ) (µ) and satisfies the functional equation E˜ p (−s ; g) = E˜ p (s ; g). (µ)

Proof. The independence of the basis Bi ( p ; 0) is clear to see. The functional equation follows from (57). Meromorphic continuation and functional equations. Let K0 (τ ) be the space of V ˜ gk) = C ∞ -functions β˜ : G → p∗C with compact support modulo 0 such that β(γ −1 ˜ τ (k) β(g) for all γ ∈ 0 and k ∈ K . Theorem 5.7. Let L 1 be the interval on the imaginary axis defined by (25). Let 0 6 j 6 µ. Then for each β˜ ∈ K0 (τ ), on Re s > n the holomorphic function ˜ := hδ j,s G˜ s |βi ˜ has a meromorphic continuation to the domain C − L 1 . s 7→ G j (s, β) A point s0 ∈ C − L 1 with Re s0 > 0 is a pole of the meromorphic function G j (s, β) (µ) ˜ (µ) 6= 0 if and only if there exists an m ∈ N such that (w|J H (α˜ m ; e)) 6= 0, hα˜ m |βi (µ) and s02 − λ2 = −λm . In this case, the function G j (s, β) −

X m∈N (µ) λm =λ2 −s02

(µ)

(µ)

˜ 4(w|J H (α˜ m ; e))hα˜ m |βi 2 2 j+1 µ! (s0 − s )

is holomorphic at s = s0 . We have the functional equation X µ (µ) ( p+1) ˜ )|βi hE˜ p (νs ˜ − G j (s, β) ˜ = δ j,s (64) . G j (−s, β) ( p+1) 2 ν s p=0 Proof. We prove the assertions by downward-induction on j. Consider the case of j = µ. By the same argument in [Oda and Tsuzuki 2003, §§6.2, 6.3], we see that the series Gdis (s) is convergent in L20 (τ ) for any s ∈ C such that for all m, (µ) λm 6= λ2 − s 2 it gives an L20 (τ )-valued meromorphic function on C (namely, for each s0 ∈ C there exists a ∈ Z such that (s − s0 )a Gdis (s) is holomorphic around s0 ). By Proposition 5.5, Gµ (s) − Gdis (s) is the sum over p of Z (µ) E˜ p (ζ ) 1 ( p,0) Gc (s) = dζ, 0 6 p 6 µ. √ √ 4π −1 −1 R (ζ 2 − (νs( p+1) )2 )r ( p+1)

Since Re νs > Re s for Re s >√0 by Lemma 3.1, the denominator of the integrand is never zero as long as ζ ∈ −1 R and Re s > 0. Hence the same argument

348

MASAO TSUZUKI ( p,0)

just cited proves the convergence of the integral Gc (s) in L20 (τ ) not only on Re s > 3n −2r but also on the broader domain Re s > 0, and moreover the integral defines a holomorphic function on Re s > 0. Thus a meromorphic continuation of P ˜ + p hG(c p,0) (s)|βi ˜ exists. Gµ (s, β) = hGdis (s)|βi The √ next step√is to obtain the analytic continuation of Gµ (s, β) around a point s0 = −1 σ0 ∈ −1 R − L 1 . For that purpose, we consider the same problem for √ ( p,0) ˜ Put ζ0 = νs(0p+1) . Note that ζ0 ∈ −1 R. Let a, b > 0 be each integrals hGc (s)|βi. (µ) ˜ 0 6 p 6 µ, are holomorphic arbitrary numbers such that the functions hE p (ζ )|βi, √ Let C√ on the open rectangle Ra,b (ζ0 ) having the vertices a,b be √ √ ζ0 ± a ± −1b. the path which, as a point set, is a union of −1 R − [ζ0 −b −1, ζ0 +b −1] and ∂ Ra,b (ζ0 ) ∩ {Re ζ > 0}, and which rounds the point ζ0 counterclockwise. Let ( p+1) Ua,b (±s0 ) be the inverse image of Ra,b (ζ0 ) by the map s 7→ νs ; thus Ua,b (±s0 ) is an open neighborhood of {s0 , −s0 } in C − L 1 . For s ∈ Ua,b (±s0 ) ∩ {Re s > 0}, by the residue theorem, ˜ (65) hG(c p,0) (s)|βi Z (µ) (µ) ˜ ˜ √ hE˜ p (ζ )|βi hE˜ p (ζ )|βi 1 = −1 Res dζ − 2π √ ( p+1) z=νs ( p+1) 2 r 4π −1 Ca,b (ζ 2 −(νs( p+1) )2 )r ) ) (ζ 2 −(νs Z ( p+1) (µ) (µ) ˜ ˜ )|βi hE˜ p (νs hE˜ p (ζ )|βi 1 1 = dζ − δ µ,s . √ ( p+1) ( p+1) 4 4π −1 Ca,b (ζ 2 − (νs )2 )r νs using Lemma 5.8 to compute the residue. The integral in the first term of this expression is convergent even for s ∈ Ua,b (±s0 ) and defines a holomorphic function on Ua,b (±s0 ). Since the second term is also holomorphic on Ua,b (±s0 ), we ob( p,0) ˜ on a neighborhood of {s0 , −s0 }. The tain an analytic continuation of hGc (s)|βi functional equation (64) for s ∈ Ua,b (±s0 ) follows if we note that the first term in the right-hand side of the second identity of (65) is invariant under the substitution ( p+1) ( p+1) s 7→ −s and also note the equation ν−s = −νs ; see Lemma 3.1. Once the functional √ equation (64) is established on a small open set of the form Ua,b (±s0 ) with s0 ∈ −1 R − L 1 , we can use it to obtain a meromorphic continuation by defining the value Gµ (s, β) for Re s < 0 in terms of Gµ (−s, β) which is defined ˜ which is meromorphic above and the terms containing the derivative of hE(µ) (s)|βi ˜ on Re s > 0 stem only from the on C; see Proposition 5.4. The poles of hGµ (s)|βi ˜ discrete part hGdis (s)|βi, whose series expression (61) itself proves the criterion in the theorem for s0 to be a pole, as well as a statement on the behavior around the poles. This completes the proof of Theorem for j = µ. We prove the theorem for j assuming it holds for j + 1. Since G j+1 (s, β) = ( j + 1)−1

1 d G j (s, β) 2s ds


349

for Re s > n by Proposition 4.4, the function G j (s, β) should be a primitive function of 2( j + 1)s G j+1 (s, β). Fix a point √ z 0 with Re z 0 > n so that the value G j (z 0 , β) is already defined. For each s0 ∈ −1 R − L 1 , let Ua,b (±s0 ) be its neighborhood constructed above and put Da,b (±s0 ) = Ua,b (±s0 )∪{Re s > 0}. We take sufficiently (µ) ( p+1) ˜ small a and b such that hE p (νs |βi are all regular on Ua,b (±s0 ). For a path Cs connecting z 0 and s ∈ Da,b (±s0 ) inside Da,b (±s0 ), consider the integral Z ˜ (66) G(s) = 2( j + 1) ζ G j+1 (ζ, β) dζ + G j (z 0 , β). Cs

Since the residues of G j+1 (s, β) at any poles in Re s > 0 are all zero by inductionassumption and since the poles of G j+1 (s, β) in Da,b (±s0 ) are automatically in Re s > 0 by the choice of a, b, this integral is independent of the choice of the path Cs and defines a meromorphic function of s on Da,b (±s0 ). Since G˜ (z 0 ) = G j (z 0 , β) and 2( j + 1)s G j+1 (s, β) = d/ds G˜ (s) for Re s > n, we have G˜ (s) = G j (s) at least on Re s > n. Thus the integral expression (66) gives us an analytic continuation of G j (s, β) to Da,b (±s0 ). The functional equation (64) for j + 1 takes the form

with

1 d J (s) = 0, 2s ds X µ (µ) ( p+1) ˜ )|βi hE˜ p (νs ˜ ˜ J (s) = G j (−s, β) − G j (s, β) − δ j,s ( p+1) 2 νs p=0

on Da,b (±s0 ). Hence J (s) is a constant on the domain Da,b (±s0 ) on the one hand. (µ) On the other hand, by the functional equation of E˜ p (ν) (Lemma 5.6) and that of ( p+1) νs (Lemma 3.1), we have J (−s) = −J (s). Hence the constant J (s) should be zero. This establishes the functional equation (64) on Da,b (±s0 ). By defining the value of G j (s, β) for Re s < 0 by the functional equation (64), we obtain the meromorphic continuation of G j (s, β) to C−L 1 keeping (64) correct. The assertion on the poles for G j (s, β) follows from that for G j+1 (s, β) from (66). ( p)

Lemma 5.8. Let 1 6 p 6 r and put νs = νs . Let U be an open domain in C − L 1 ( p) and F(z) a holomorphic function on some open neighborhood of {νs | s ∈ U }. Then for each j > 1 we have F(z) F(νs ) Resz=νs 2 = δ , s ∈ U. j−1,s (z − νs2 ) j 2νs 6. Green currents For Re s > n, the currents G s := (cr µ µ!)−1 P(φs ) and 9s = (cr µ µ!)−1 P(ψs ) are of (µ, µ)-type and of (r, r )-type respectively. In this section we study some of the properties of the currents G s and 9s using the knowledge of the function G˜ s

350

MASAO TSUZUKI

obtained in the previous section. We put the Kähler form ω on 0\G/K such that ˜ ω(g) = ω for all g ∈ G. The metric on 0\G/K corresponding to ω defines the Laplacian 4, the Lefschetz operator and its adjoint 3 acting on the space of forms and currents on 0\G/K . Currents defined by modular cycles. Denote by D the image of the map from 0 H \H/K H to 0\G/K induced by the natural holomorphic inclusion of H/K H into G/K . Then D, a closed complex analytic subset of 0\G/K , defines an (r, r )current δ D on 0\G/K by the integration Z hδ D , αi = j ∗ α, α ∈ Ac (0\G/K ). Dns

Here j : D ,→ 0\G/K is the natural inclusion and Dns is the smooth locus of D. Since δ D is closed, it defines a cycle on 0\G/K of real codimension 2r [Griffiths and Harris 1978, p. 32–33]. Proposition 6.1. For α ∈ Ac (0\G/K ), we have (67)

hδ D , ∗αi = (∗vol H |J H (α˜ ; e)),

(68)

h3δ D , ∗αi = (3(∗vol H )|J H (α˜ ; e)).

Proof. We give a proof assuming the natural map p : 0 H \H/K H → 0\G/K is one-to-one. (The general case is similar.) For any β ∈ Ac (0\G/K ), we have Z Z ∗ ˙ hδ D , βi = p β= ( f (h)|vol H ) d h, 0 H \H/K H

0 H \H

noting that kvol H k = 1. Here f : H → (h∩p)∗C is the function on H corresponding to p ∗ β which is determined by a formula similar to (48). Put β = ∗α. Equation (67) holds since V

˜ ( f (h)|vol H ) = (β(h)|vol ˜ = (∗vol H |α(h)). ˜ H ) = (vol H | ∗ α(h)) Thus (68) follows from (67), since h3δ D , βi = hδ D , 3βi by definition.

Differential equations. First we show that G s and 9s satisfy some differential equations. Theorem 6.2. Let Re s > n. Then (69) (70) (71)

(4 + s 2 − λ2 )G s = −43δ D , √ 4 9s = (λ2 − s 2 )(9s − 2 −1δ D ), √ ¯ s = 9s − 2 −1 δ D . ∂ ∂G


351

Proof. It suffices to prove these formulas for s with Re s > 3n − 2r because they depend on s holomorphically; see Proposition 4.4. Since R α˜ = −(4α)∼ for α ∈ A(0\G/K ) by Kuga’s formula, equation (69) follows from Theorem 4.5, the first equality in (7) and (68). To prove (71), take an arbitrary form α ∈ Ac (0\G/K ). First by definition and then by an application of (50), ¯ s , ∗αi = −hG s , ∂∂ ¯ ∗ αi = hG s , ∗∂ ∗ ∂¯ ∗ αi = hG s , ∗∂¯ ∗ ∂ ∗ αi h∂ ∂G Z ∞ 1 = %(t) (φs (at )|J H ( f ; at )) dt, cr µ µ! 0

(72)

with f (g) = (∂¯ ∗ ∂ ∗ α)∼ (g). Since n−1 X

(∂¯ ∗ ∂ ∗ α)∼ (g) = −

α,β=0

e(ωα )∗ e(ωβ )∗ R X α X β α(g), ˜

we have J H ( f ; at ) = −

(73)

n−1 X α,β=0

e(ωα )∗ e(ωβ )∗ R X α X β J H (α˜ ; at ),

using Proposition 4.3(1). Inserting the formulas in Lemma 2.1 to the right-hand side of (73), we obtain J H ( f ; at ) = Et J H ( f ; at ) with Et the differential operator on t > 0 given by Et = −

1 e∗ (ω0 ∧ ω0 ) d 2 ˜ coth t) d + (A˜ tanh t + B 2 4 dt 2 dt 2 ˜ ˜ ˜ ˜ − coth2 t − P ˜ +R ˜ −−R ˜ +P ˜ −, − P+ P− tanh t − R+ R

where ˜+ = P ˜+= R

n−r X

e (ωi )τ (Z i ), ∗

˜− = P

n−r X

i=1

i=1

n−1 X

n−1 X

e∗ (ω j )τ (Z j ),

˜−= R

j=n−r +1

e∗ (ωi )τ (Z i ), e∗ (ω j )τ (Z j ),

j=n−r +1

√ ˜ + − e∗ (ω0 )P ˜ −, A˜ = e∗ (2 −1ω H ) − 12 e∗ (ω0 ∧ ω0 ) + e∗ (ω0 )P √ ˜ + − e∗ (ω0 )R ˜ −. ˜ = e∗ ( 2 −1 η ) − 1 e∗ (ω0 ∧ ω0 ) + e∗ (ω0 )R B 2 Let 0 < ε < R. Integration by parts yields R

Z (74)

ε

%(t) (φs (at )|J H ( f ; at )) dt = 8(R) − 8(ε) +

R

Z ε

%(t) (E∗t φs (at )|J H ( f ; at )) dt,

352

MASAO TSUZUKI

with (75) 8(t) = − 14 %(t) ω0 ∧ ω0 ∧ φs (at )|F 0 (t) d + 41 %0 (t) ω0 ∧ ω0 ∧ φs (at )|F(t) + 41 %(t) ω0 ∧ ω0 ∧ φs (at )|F(t) dt ∗ ∗ 1 ˜ ˜ + 2 %(t) (A tanh t + B coth t)φs (at )|F(t) . Here E∗t is the formal adjoint operator of Et , uniquely determined by the relation Z ∞ Z ∞ (E∗t a(t)|b(t)) %(t) dt = (76) (a(t)|Et b(t)) %(t) dt 0

0

for any compactly supported functions a(t) and b(t) on (0, ∞). We compute the limit of 8(ε) as ε going zero. By (40), cr ˜ ∗ )w|F(0) 1 + O(s ; ε, ε log ε) . (77) 8(ε) = (e(ω0 ∧ ω0 ) + 2B 2 V

p∗C M -valued

Using the formulas in Lemma 2.1 and the relation τ (Z )∗ = −τ (Z ) (Z ∈ kC ), we compute to obtain √ ˜ ∗+ e(ω0 ) = −e(2 −1η + µ ω0 ∧ ω0 ) + e(ω0 )R+ , R √ ˜ ∗− e(ω0 ) = e(2 −1η + µ ω0 ∧ ω0 ) + e(ω0 )R− . R ˜ ∗+ e(ω0 )w = Using these formulas, (6) and the relations R± w = 0, we obtain R ˜ ∗− e(ω0 )w = 0. Hence R √ ˜ ∗− e(ω0 )w ˜ ∗+ e(ω0 )w − R ˜ ∗ w = e(2 −1η)w − 1 e(ω0 ∧ ω0 )w + R B 2 √ = 2 −1η ∧ w − 21 ω0 ∧ ω0 ∧ w. From this formula and (77), √ √ (78) lim 8(ε) = 2 −1cr (η ∧ w|F(0)) = 2 −1cr µ µ!(∗vol H |F(0)) ε→+0

using (6) to obtain the second equality. We compute the limit of 8(R) as R tends to infinity by means of (41) noting Lemma 3.1. The result is, when Re s > n, (79)

lim 8(R) = 0.

R→+∞

Putting (72), (74), (78) and (79) together and using (67), we finally obtain Z ∞ √ 1 ¯ h∂ ∂G s , ∗αi = −2 −1hδ D , ∗αi + %(t) (E∗t φs (at )|J H ( f ; at )) dt. cr µ µ! 0 To complete the proof of (71), we have only to prove that E∗t φs (at ) = ψs (at ) for t > 0. Since ψs (at ) = E˜ t φs (at ) with E˜ t the differential operator in t given by (24)


353

in the z-coordinate, it suffices to show E∗t = E˜ t . For that purpose, we show that Z ∞ Z ∞ E˜ t a(t)|b(t) %(t) dt = (E∗t a(t)|b(t)) %(t) dt 0

0

for arbitrary compactly supported pC -valued C ∞ -functions a(t) and b(t) on (0, ∞). By the decomposition G = HAK , we can extend the functions a(t) and b(t) to smooth functions a(g) and b(g) belonging to Cτ∞ by the formula V

a(hat k) = τ (k)−1 a(t),

∗ M

and b(hat k) = τ (k)−1 b(t),

h ∈ H, t > 0, k ∈ K .

By Lemma 4.1, (76) and the definitions of Et and E˜ t , we obtain X n−1

Z H \G

Z H \G

α,β=0

Z e(ωβ )e(ωα )R X α X β a(g) b(g) d g˙ =

Z n−1 X ∗ ∗ a(g) e(ωα ) e(ωβ ) R X α X β b(g) d g˙ = α,β=0

∞

(E˜ t a(t)|b(t)) %(t) dt, 0 ∞ 0

(E∗t a(t)|b(t)) %(t) dt.

The left-hand sides of these two formulas are easily seen to be identical, since Z Z ˙ a1 , a2 ∈ Cτ∞ , X ∈ gC . (a1 (g)|R X a2 (g)) d g˙ = − (R X a1 (g)|a2 (g)) d g, H \G

H \G

We now deduce (70) from (69) and (71). That the current δ D is real and closed ¯ D = 0. Hence 4δ D = d d∗ δ D = (∂+∂)(∂ ¯ ∗ +∂¯ ∗ )δ D . Since implies that ∂δ D = 0 and ∂δ √ √ ∗ ¯ and ∂¯ ∗ = − −1[3, ∂]; 0\G/K is a Kähler manifold, we have ∂ = −1[3, ∂] see [Wells 1980, (4.5), Corollary 4.10, p. 193]. Using these equalities, we compute √ √ √ ¯ ¯ D = −1(∂ + ∂)(∂ ¯ − ∂)3δ ¯ ¯ D ∂ − ∂])δ 4δ D = −1(∂ + ∂)(−[3, D = −2 −1∂ ∂3δ to obtain (80)

√ ¯ D = 0. 4δ D + 2 −1∂ ∂3δ

√ ¯ s + 2 −1δ D by (71), we have Since 9s = ∂ ∂G √ √ ¯ s + 2 −1δ D ) = ∂ ∂¯ 4 G s + 2 −1 4 δ D 49s = 4(∂ ∂G √ ¯ 2 − s 2 )G s − 43δ D ) + 2 −1 4 δ D = ∂ ∂((λ √ ¯ s − 4∂ ∂3δ ¯ D + 2 −1 4 δ D = (λ2 − s 2 )∂ ∂G √ √ √ ¯ D) = (λ2 − s 2 )(9s − 2 −1δ D ) + 2 −1(4δ D + 2 −1∂ ∂3δ √ = (λ2 − s 2 )(9s − 2 −1δ D ), using (69) for the third equality, (71) for the fifth and (80) for the last.

354

MASAO TSUZUKI p,q

Main theorem. Let A(2) (0\G/K ) be the Hilbert space of the measurable ( p, q)R forms on 0\G/K with the finite L 2 -norm kαk := ( 0\G/K α ∧ ∗α)1/2 . For each p,q c ∈ C, let A(2) (0\G/K ; c) be the c-eigenspace of the Laplacian 4 acting on p,q A(2) (0\G/K ). In particular, p,q

p,q

H(2) (0\G/K ) := A(2) (0\G/K ; 0) (µ)

is the space of the harmonic L 2 -forms of ( p, q)-type. For each p, let E p (ν) be (µ) the C ∞ -form of (µ, µ)-type on 0\G/K corresponding to the function E˜ p (ν) on G defined by (63). Then Theorem 5.7 immediately gives us the following theorem. Theorem 6.3. There exists a meromorphic family of (µ, µ)-currents G s , with s ∈ C − L 1 , on 0\G/K with the following properties. (1) For s ∈ C with Re s > n, the family is given by Z ∞ 1 hG s , ∗αi = %(t) (φs (at )|J H (α˜ ; at )) dt, (r − 1)π r 0

α ∈ Ac (0\G/K ).

(2) A point s0 ∈ C − L 1 with Re s > 0 is a pole of G s if and only if there exists an −1,r −1 (0\G/K ; (n − 2r + 2)2 − s02 ) such that L 2 -form α ∈ Ar(2) Z j ∗ ∗ (ω ∧ α) 6= 0. D

In this case s0 is a simple pole with residue Z 2 X Ress=s0 G s = j ∗ ∗ (ω ∧ α m ) · αm . s0 m D −1,r −1 (0\G/K ; (n − 2r + 2)2 − s02 ). where {αm } is an orthonormal basis of Ar(2)

(3) The following functional equation holds: G −s − G s =

r −1 (r −1) ( p+1) X E p (νs ) ( p+1)

p=0

2νs

,

s ∈ C − L 1.

Theorem 6.4. There exists a meromorphic family of (r, r )-currents 9s s ∈ C − L 1 , on 0\G/K such that for s ∈ C with Re s > n, the current 9s is given by Z ∞ 1 h9s , ∗αi = %(t) (ψs (at )|J H (α˜ ; at )) dt, α ∈ Ac (0\G/K ). (r − 1)π r 0 Moreover, 9s is holomorphic at s = n − 2r + 2. Proof. The meromorphic continuation of 9s follows from the differential equation (71) and the meromorphicity of G s . Let β be the residue of G s at s = λ. Then


355

by Theorem 6.3(2), β is a harmonic L 2 -form. Moreover dβ and d∗ β are also L 2 forms due to the fact that for any D ∈ gC , the derivative R D β˜ is square-integrable on 0\G. Then we can conclude that dβ = 0 and d∗ β = 0 using the identity 0 = k 4 βk2 = kdβk2 + kd∗ βk2 . ¯ s at its possible simple pole s = λ is Since the residue of the function s 7→ ∂ ∂G √ −1 ¯ =2 ¯ s is regular at s = λ. By (71), 9s is ∂ ∂β −1dc dβ = 0, the function ∂ ∂G also regular at s = λ. Definition. We define the (r −1, r −1)-current G on 0\G/K to be the quarter of the constant term of the Laurent expansion of G s at s = λ, Namely, if {αm } is any −1,r −1 orthonormal basis of Hr(2) (0\G/K ), we put XZ 2 αm (x) 1 ∗ . G(x) = lim G s (x) − j ∗ (ω ∧ α m ) 4 s→λ n − 2r + 2 m D s − (n − 2r + 2) Theorem 6.5. We have the equation √

ddc G =

−1 2 9n−2r +2 + δ D ,

49n−2r +2 = 0.

The current 9n−2r +2 is represented by an element of Ar,r (0\G/K ). Proof. Since 9s is regular at s = λ, the differential equation (70) gives us 49λ = 0. The current 9λ , which is annihilated by the elliptic differential operator 4 on 0\G/K , is then a C ∞ -form by the elliptic regularity theorem. By comparing the constant terms of the Laurent expansion at s = λ of both sides of the identity (71), we obtain the first equation in the theorem. 7. Square-integrability of 9λ In this section we prove the square-integrability of 9λ . To establish it we need the spectral expansion of the functions ˜ j,s ((s 2 − λ2 )−1 ψs )). e s ) := (cr µ µ!)−1 P(δ δ j,s ((s 2 − λ2 )−1 9 Lemma 7.1. Let cα (s) (0 6 α 6 µ−1) be the functions appearing as coefficients in the asymptotic formula (44). For each α there exists an even polynomial function c˜ α (s) with √ degree no more than 2α such that cα (s) = (s 2 − λ2 )˜cα (s). We have c˜ 0 (s) = −( −1/2) µ! ∗ vol H , independent of s. For each 0 6 j 6 µ, we have ψs (at ) w ˜ µ− j (81) δ j,s 2 = z µ− z = tanh 2 t, j (1 + O(s ; z, z log z)), s − λ2 with w˜ α := (δ α,s c˜ α )(0).

356

MASAO TSUZUKI

Proof. For f ∈ K0 (τ ), consider the integral Z e s (g)|R+λ2 −¯s 2 f (g)) d g. I( f ) = ((s 2 − λ2 )−1 9 ˙ 0\G

√ By (70) and (67), I ( f ) = −2 −1(∗vol H |J H ( f ; e)) on the one hand. On the other hand, we can compute I ( f ) in a way similar to that in the proof of Theorem 4.5 to obtain 4 I ( f ) = (s 2 − λ2 )−1 (c0 (s)|J H ( f ; e)). µ! Comparing these two expressions for I ( f ), we get (82)

√ 4 −2 −1(∗vol H |J H ( f ; e)) = (s 2 − λ2 )−1 (c0 (s)|J H ( f ; e)). µ!

By choosing f suitably, we can arrange for the value of J H ( f ; e) to be any element V of ( r,r p∗C ) M . Hence the relation (82) implies (83)

c0 (s) = −

√

−1 2 µ!

∗ vol H (s 2 − λ2 ).

Since the Schmid operators ∇± commute with the Casimir operator , (26) implies that ψs is also a C ∞ -solution of (22). Hence the same argument as in the proof of Theorem 3.14 yields that the coefficients cα (s) must obey the same recurrence V relation (43) as aα (s). Since the operator S0 −α(µ−α) is invertible on r,r p∗C M , the recurrence relation (43) with the initial condition (83) implies the first and the second assertions in the lemma. The last assertion follows from the expression cα (s) = (s 2 − λ2 )˜cα (s) just obtained. Theorem 7.2. Let 0 6 j 6 µ be an integer. Suppose Re s > 3n − 2r . Then for any f ∈ Mδ with δ ∈ (2r n −1 , 1), we have Z √ e s )(g)|R(+λ2 −¯s 2 ) j+1 f (g)) d g˙ = −2 −1 (∗vol H |J H ( f ; e)). (δ j,s ((s 2 −λ2 )−1 9 0\G

Proof. The proof is analogous to that of Theorem 4.5. We use (81).

Theorem 7.3. Let Re s > 3n − 2r . There exists ε > 0 such that the function (r ) e s ) belongs to the space L2+ε δ µ,s ((s 2 − λ2 )−1 9 0 (τ ) . The spectral expansion of 2 2 −1 e δ µ,s ((s − λ ) 9 s ) is δ µ,s

es 9 s 2 − λ2

= Fdis (s) +

r X p=0

F(c p) (s),


357

with Fdis (s) =

√ ∞ X −2 −1 (∗vol H |J H (α˜ m(r ) ; e)) ) 2 r (λ2 − λ(r m −s )

m=0

Fc( p) (s)

1 = √ 4π −1

α˜ m(r ) ,

√ h X X −2 −1(∗vol H |J H (Ei (ζ ; u); e))

Z

( p+1) 2 r ) )

√ −1 R i=1

(ζ 2 − (νs

u

Ei (ζ ; u) dζ,

where the inner summation is over u ∈ Bi(r ) ( p; 0) and both summations are convergent in L20 (τ )(r ) .

Proof. Like that of Theorem 5.3. We use Theorem 7.2.

Theorem 7.4. Let L 1 be the interval on the imaginary axis defined by (25). Let 0 6 j 6 µ. Then for each β˜ ∈ K0 (τ ) the holomorphic function ˜ := hδ j,s (s 2 − λ2 )−1 9 ˜ e s |βi s 7→ F j (s, β) on Re s > n has a meromorphic continuation to the domain C − L 1 . A point s0 ∈ C − L 1 with Re s0 > 0 is a pole of the meromorphic function F j (s, β) if and only if there exists an m ∈ N such that (∗vol H |J H (α˜ m(r ) ; e)) 6= 0,

˜ 6= 0, hα˜ m(r ) |βi

) and s02 − λ2 = −λ(r m . In this case, the function

F j (s, β) −

X m∈N ) 2 2 λ(r m =λ −s0

√ ˜ 2 −1(∗vol H |J H (α˜ m(r ) ; e))hα˜ m(r ) |βi 2 2 j+1 (s0 − s )

is holomorphic at s = s0 . We have the functional equation (84)

˜ − F j (s, β) ˜ = δ j,s F j (−s, β)

X r ( p+1) ˜ hE˜ (rp ) (νs )|βi ( p+1)

p=0

2 νs

,

where, for g ∈ G, h √ X (r ) ˜ E p (ν ; g) := −2 −1

X

(∗vol H |J H (Ei (−¯ν ; u); e)) Ei (ν ; u ; g).

i=1 u∈B(r ) ( p ;0) i

Proof. The proof is the same as that of Theorem 5.7. The statement for j = µ follows from Theorem 7.3. Then we use induction to show the statement for j smaller than µ.

358

MASAO TSUZUKI

Some properties of the current 9s . Theorem 7.5. (1) A point s0 ∈ C − L 1 with Re s0 > 0 and s0 6= n − 2r + 2 is a pole of the current 9s if and only if there exists an L 2 -form 2 2 α ∈ Ar,r (2) 0\G/K ; (n − 2r + 2) − s0 such that

Z

j ∗ ∗α 6= 0. D

In this case s0 is a simple pole with residue √ Z −1(s02 − (n − 2r + 2)2 ) X ∗ Ress=s0 9s = j ∗ α m · αm , s0 D m 2 2 where {αm } is an orthonormal basis of Ar,r (2) (0\G/K ; (n − 2r + 2) − s0 ).

(2) We have

√ X 9n−2r +2 = 2 −1 m

Z

j ∗ β m · βm , ∗

D

with {βm } an orthonormal basis of Hr,r (2) (0\G/K ). In particular, the current r,r 9n−2r +2 is in H(2) (0\G/K ).

Proof. This is a corollary of Theorem 7.4.

By Theorem 6.5, the fundamental class [δ D ] ∈ H (0\G/K ; C) of D has the harmonic L 2 -representative 9n−2r +2 . r,r

References [Borel and Wallach 1980] A. Borel and N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies 94, Princeton University Press, Princeton, NJ, 1980. MR 83c:22018 Zbl 0443.22010 [Gillet and Soulé 1990] H. Gillet and C. Soulé, “Arithmetic intersection theory”, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174 (1991). MR 92d:14016 Zbl 0741.14012 [Gon and Tsuzuki 2002] Y. Gon and M. Tsuzuki, “The resolvent trace formula for rank one Lie groups”, Asian J. Math. 6:2 (2002), 227–252. MR 2003j:22011 Zbl 1026.22010 [Griffiths and Harris 1978] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 95d:14001 Zbl 0408.14001 [Heckman and Schlichtkrull 1994] G. Heckman and H. Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics 16, Academic Press, San Diego, CA, 1994. MR 96j:22019 Zbl 0836.43001 [Hejhal 1983] D. A. Hejhal, The Selberg trace formula for PSL(2, R). Vol. 2, Lecture Notes in Mathematics 1001, Springer, Berlin, 1983. MR 86e:11040 Zbl 0543.10020 [Magnus et al. 1966] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer, New York, 1966. MR 38 #1291 Zbl 0143.08502


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[Miatello and Wallach 1989] R. Miatello and N. R. Wallach, “Automorphic forms constructed from Whittaker vectors”, J. Funct. Anal. 86:2 (1989), 411–487. MR 91b:11065 Zbl 0692.10029 [Miatello and Wallach 1992] R. Miatello and N. R. Wallach, “The resolvent of the Laplacian on locally symmetric spaces”, J. Diff. Geom. 36:3 (1992), 663–698. MR 93i:58160 Zbl 0766.53044 [Oda and Tsuzuki 2003] T. Oda and M. Tsuzuki, “Automorphic Green functions associated with the secondary spherical functions”, Publ. Res. Inst. Math. Sci. 39:3 (2003), 451–533. MR 2004f:11046 Zbl 1044.11033 [Wells 1980] R. O. Wells, Jr., Differential analysis on complex manifolds, Graduate Texts in Mathematics 65, Springer, New York, 1980. MR 83f:58001 Zbl 0435.32004 Received December 15, 2004. Revised October 16, 2005. M ASAO T SUZUKI D EPARTMENT OF M ATHEMATICS S OPHIA U NIVERSITY K IOI - CHO 7-1 C HIYODA - KU T OKYO , 102-8554 JAPAN [email protected]


EXISTENCE OF SOLUTIONS AND REGULARITY NEAR THE CHARACTERISTIC BOUNDARY FOR SUB-LAPLACIAN EQUATIONS ON CARNOT GROUPS D IMITER VASSILEV We prove that the best constant in the Folland–Stein embedding theorem on Carnot groups is achieved. This implies the existence of a positive solution of the Yamabe-type equation on Carnot groups. The second goal of the paper is to show a certain regularity of the Green’s function and solutions of the Yamabe equation involving the sub-Laplacian near the characteristic boundary of a domain in the considered groups.

1. Introduction In this paper we consider problems of existence and regularity of solutions to a nonlinear Dirichlet problem involving sub-Laplacians on Carnot groups. The main motivation comes from the classical Yamabe problem, the question of determining the best constant in the Sobolev embedding inequality, and their CR counterparts. A Carnot group is a simply connected and connected Lie group G, whose Lie L algebra g admits a stratification g = rj=1 Vj with [V1 , Vj ] = Vj+1

for 1 ≤ j < r,

[V1 , Vr ] = {0}.

Pr

Let Q = j=1 j dim Vj be the homogeneous dimension. Our starting point is this embedding result of Folland and Stein, [Folland 1975]: For any p ∈ (1, Q) there exists S p = S p (G) > 0 such that, for u ∈ C0∞ (G), Z 1/ p∗ Z 1/ p p∗ p (1-1) |u| d H ≤ Sp |X u| d H . G

G

Pm 2 1/2 , where Here, the horizontal gradient |X u| is defined as |X u| = j=1 (X j u) X = {X 1 ,. . . ,X m } is a basis of V1 , p ∗ = p Q/(Q − p), and d H is a fixed Haar measure on G. Unlike the Euclidean case [Talenti 1976; Aubin 1976b], the value of the best possible constant or the nonnegative functions for which it is achieved is unknown. MSC2000: 35J70. Keywords: subelliptic regularity, Sobolev embedding, Carnot groups. 361

362

DIMITER VASSILEV

In the particular case when p = 2 and G = Rn , this problem is related to the Yamabe problem ([Trudinger 1968; Aubin 1976a; Schoen 1984], see also the survey article [Lee and Parker 1987]); in the case of the Heisenberg group, it is related to the CR Yamabe problem. The case of odd-dimensional spheres is equivalent to the problem of determining the best constant in the L 2 Folland–Stein inequality on the Heisenberg group, and the functions for which it is achieved. The solution of this problem is instrumental for solving the general case, which is complete after the works of Jerison and Lee [1984; 1987; 1988; 1989], and Gamara and Yacoub [Gamara 2001; Gamara and Yacoub 2001]. We note that, when p = 2, the Yamabe equation is the Euler–Lagrange equation satisfied by the extremals of the naturally associated variational problem. It is also of interest to study the Yamabe equation on sets different from the whole group, especially in connection with certain blow-up arguments that appear, for example, in questions of existence of solutions on domains with a nontrivial topology [Bahri and Coron 1988; Brezis 1986; Citti and Uguzzoni 2001]. We consider similar problems in a general Carnot group. The method of concentration compactness of P. L. Lions can be used, and we show in Section 3 that there exists a best constant in the Folland–Stein inequality. It is achieved on the ˚ 1, p (G), which is the closure of C ∞ (G) with respect to the norm space D 0 Z kukD˚ 1, p (G) =

p

1/ p

|X u| d H

.

G

This method does not allow an explicit determination of the best constant or the functions for which it is achieved. The problem can be formulated as a variational problem. We consider the case p = 2 in detail. The Euler–Lagrange equation of the nonnegative extremals leads to the Yamabe-type equation ∗ −1

Lu = −u 2

(1-2)

,

Pm where Lu = j=1 X j2 u. While it is relatively easy to see that, in any domain, weak solutions to this equation are bounded, further regularity is based on intricate subelliptic estimates. In Section 4 we show that any weak solution of the equation (1-3)

Lpu =

n X

∗ X j |X u| p−2 X j u = −|u| p −2 u

in G.

j=1

is a bounded function. We shall present the proof in a somewhat more general case. There are several similar results in the Riemannian case. Yamabe [1960], proved the boundedness for solutions of the Yamabe problem on a manifold without boundary; see also [Trudinger 1968]. For the ordinary Laplacian

EXISTENCE AND REGULARITY FOR SUB-LAPLACIAN EQUATIONS

363

in a bounded domain, Brézis and Kato [1979] established a result similar to part (1) of Theorem 4.1; for part (2), see [Brézis and Nirenberg 1983]. The proofs rely on a suitable modification of the test-function and truncation ideas introduced in Serrin’s seminal paper [1964] and in [Moser 1961]. We note that, in all these results, u is assumed from the space L p (), since this is part of the definition of ˚ 1, p () that the considered Sobolev spaces. This is not true in the Sobolev spaces D p we consider, since we only include the L norm of the horizontal gradient in our definition. Therefore, the results here are not exactly the same, besides our working on a Carnot group. Subsequently, Serrin’s ideas were generalized to the subelliptic setting in [Capogna et al. 1993] and, in different forms, also in [Holopainen 1992; Holopainen and Rickman 1992; Lanconelli and Uguzzoni 1998; Xu 1990]. The rest of the paper concerns the regularity of solutions on bounded domains. In Section 5 we show that, under certain geometric conditions on the boundary of the C ∞ connected bounded open set ⊂ G, one can prove boundedness of the horizontal gradient of the Green’s function. We shall also obtain estimates along other vector fields. Similar results hold near the boundary of the domain for weak nonnegative solutions of the following Yamabe-type equation (1-4)

∗ −1

Lu = −u 2

,

˚ 1,2 (), u∈D

u ≥ 0.

In order to prove such estimates, we impose some geometrical conditions on the boundary. In particular, we point to the “convexity condition” [Garofalo and Vassilev 2000], assumed to hold globally whenever we are working with nonlinear equations, unless we are on the Heisenberg group. The reason for such an assumption is that, at present, there is no proof of the boundary Schauder estimates in Lipschitz spaces, or of 0 2,α regularity near the points from the noncharacteristic portion of the boundary. Assuming the global validity of the convexity condition allows us to avoid the use of Schauder estimates or extra a priori regularity assumptions. However, by using the Lipschitz Schauder theory near the noncharacteristic boundary for domains on the Heisenberg group [Jerison 1981a], we present in Section 5E the argument in the local setting, with the convexity assumption holding only near the characteristic boundary. An example in [Jerison 1981b] implies that the estimates we obtain at the characteristic boundary fail without some assumption on the boundary. However, the convexity condition is clearly not necessary for estimates of the type considered. Boundedness of the horizontal gradient has been established for more general domains, satisfying a uniform outer-ball condition [Capogna et al. 1998]. Verifying such a condition is in general hard. A result as in the C 1,1 -boundary case in Rn is not true. Also, the horizontal gradient involves only differentiation along vectors from the first layer. We give estimates along vector fields involving differentiation along vectors from other layers; in particular, we include the radial vector field.

364

DIMITER VASSILEV

Lastly, in [Garofalo and Vassilev 2000] it was required that the domain be uniformly star-like near the characteristic boundary, which is unnecessary, as we will show. 2. Preliminaries and notation We introduce the relevant definitions and state some results which will be needed in the sequel. Consider the Lie algebra g = ⊕rj=1 Vj of G. We assume that there is a scalar product on g, with respect to which the Vj ’s are mutually orthogonal. Let X = {X 1 ,. . . ,X m } be a basis of V1 , and continue to denote by X the corresponding system of sections on G. The sub-Laplacian associated with X is the second-order partial differential operator on G given by L=−

m X

X j∗ X j =

j=1

m X

X j2

j=1

(recall that X j∗ = −X j in a Carnot group; see [Folland 1975]). From the assumption on the Lie algebra, one immediately sees that the system X satisfies the well-known finite-rank condition, and therefore that the operator L is hypoelliptic, thanks to Hörmander’s theorem [1967]. However, it fails to be elliptic, and the loss of regularity is measured by the step r of the stratification of g. For a function u on G, set P 1/2 m |X u| = (X j u)2 . j=1

˚ 1, p (G) as the closure of C ∞ (G) with respect to the norm For 1 ≤ p < Q, define D 0 Z (2-1)

kukD˚ 1, p (G) =

p

|X u| d H

1/ p

.

G

We define the Sobolev exponent relative to p as the number p ∗ = p Q/(Q − p), where Q is the homogeneous dimension defined below. The relevance of such a number is emphasized by the Folland–Stein inequality (1-1). In any Carnot group, the exponential mapping exp : g → G is an analytic diffeomorphism. We use it to define analytic maps ξi : G → Vi , i = 1, . . . , r , through the equation g = exp ξ(g), where ξ(g) = ξ1 (g) + · · · + ξr (g). With m = dim(V1 ), the coordinates of ξ ’s projection in the basis X 1 ,. . . ,X m will be denoted by x1 = x1 (g), . . . , xm = xm (g), that is, (2-2)

xj (g) = hξ(g), X j i,

j = 1, . . . , m.

We set x = x(g) = (x1 , . . . , xm ) ∈ Rm . Later we will need to exploit the properties of the exponential coordinates in the second layer of the stratification of g. We


365

thus fix an orthonormal basis Y1 , . . . , Yk of V2 and, similarly to (2-2), define the exponential coordinates in the second layer V2 of a point g ∈ G by setting (2-3)

yi (g) = hξ(g), Yi i,

i = 1 . . . k,

and y = (y1 , . . . , yk ) ∈ Rk . Every Carnot group is naturally equipped with a family of nonisotropic dilations defined by (2-4)

δλ (g) = exp ◦ 1λ ◦ exp−1 (g),

g ∈ G,

where exp : g → G is the exponential map and 1λ : g → g is defined by 1λ (ξ1 + · · · + ξr ) = λξ1 + · · · + λr ξr . P The topological dimension of G is N = rj=1 dim Vj , whereas the homogeneous dimension of G, attached to the automorphisms {δλ }λ>0 , is given by Q=

r X

j dim Vj .

j=1

One has d H (δλ (g)) = λ Q d H (g), so that, with respect to the group dilations, the number Q plays the role of a dimension. Let Z be the infinitesimal generator of the one-parameter group of nonisotropic dilations {δλ }λ>0 . Such a vector field is characterized by the property that a functionu : G → R is homogeneous of degree s with respect to {δλ }λ>0 — that is, u δλ (x) = λs u(x) for every x ∈ G — if and only if Z u = su. The Euclidean distance to the origin | · | on g induces a homogeneous norm | · |g on g, and (via the exponential map) a norm on the group G, in the following way (see also [Folland 1975]): First, for ξ ∈ g with ξ = ξ1 + · · · + ξr , ξi ∈ Vi , set P 2r ! r (2-5) |ξ |g = |ξi |2r !/i ; i=1

then define |g| G = |ξ |g if g = exp ξ . Such a norm on G can be used to define a pseudodistance on G: (2-6)

ρ(g, h) = |h −1 g| G .

The pseudodistance ρ is equivalent to the Carnot–Carathéodory distance d( ·, · ) generated by the system X , that is, there exists a constant C = C(G) > 0 such that (2-7)

Cρ(g, h) ≤ d(g, h) ≤ C −1ρ(g, h),

g, h ∈ G,

see [Nagel et al. 1985]. We will almost exclusively work with the distance d, except in a few situations where we will find more convenient to use (2-6). The

366

DIMITER VASSILEV

Carnot–Carathéodory balls are defined in the obvious way, B R (x) ≡ B(x, R) = {y ∈ G | d(x, y) < R}. By left-translation and dilation it is easy to see that the Haar measure of B(x, R) P is proportional to R Q , where Q = ri=1 i dim Vi is the homogeneous dimension of G. One has, for every f, g, h ∈ G and any λ > 0, (2-8) d(g f, gh) = d( f, h), d δλ (g), δλ (h) = λd(g, h). We also recall the Baker–Campbell–Hausdorff formula; see, for example, [Hörmander 1967]: ξ, η ∈ g, (2-9) exp ξ exp η = exp ξ + η + 21 [ξ, η] + · · · , where the dots indicate a linear combination of terms of order three and higher, which is finite due to the nilpotency of G. By definition, the order of an element in Vj is j. We next list some results which play an important role in this paper. To state these, we recall that, given a bounded open set D ⊂ G and a function ϕ ∈ C(∂ D), the Dirichlet problem for D and a sub-Laplacian L consists in finding a solution to Lw = 0 in D which takes value ϕ on the boundary. Theorem 2.1 (Bony’s maximum principle [Bony 1969]). Let D ⊂ G be a connected bounded open set, and ϕ ∈ C(∂ D). There exists a unique L-harmonic function HϕD that solves the Dirichlet problem in the sense of Perron–Wiener–Brelot. Moreover, HϕD satisfies sup |HϕD | ≤ sup |ϕ|. D

∂D

Theorem 2.2 (Schauder-type interior estimates [Danielli and Garofalo 1998]). Let D ⊂ G be an open set, and suppose that w is L-harmonic in D. For every g ∈ D ¯ and r > 0 with B(g, r ) ⊂ D, one has C |X j1 X j2 . . . X js w(g)| ≤ s max |w|, r B(g,r ¯ ) for s ∈ N, j1 , . . . , js ∈ {1, . . . , m}, and some constant C = C(G, s) > 0. To state the next result, we introduce a definition. Given an open set D ⊂ G, we denote by L1,∞ (D) the space of those distributions u ∈ L ∞ (D) such that X u ∈ L ∞ (D), endowed with the natural norm. Theorem 2.3 (L ∞ Poincaré inequality [Garofalo and Nhieu 1998]). For a Carnot group G, there exists C = C(G) > 0 such that, if u ∈ L1,∞ (B(g0 , 3R)), then u can ¯ 0 , R) so as to satisfy be modified on a set of measure zero in B(g |u(g) − u(h)| ≤ C d(g, h)kukL1,∞ (B(g0 ,3R))


367

¯ 0 , R). If furthermore u ∈ C ∞ (B(g0 , 3R)), then only the L ∞ for every g, h ∈ B(g norm of X u suffices in the right-hand side of the previous inequality. We note explicitly that the theorem asserts that every function u ∈ L1,∞ (B(g0 , 3R)) has a representative which is Lipschitz-continuous in B(g0 , R) with respect to the Carnot–Carathéodory distance d. The reverse implication also holds; see [Garofalo and Nhieu 1998]. Finally, we recall a characterization of the nonisotropic Lipschitz spaces on Carnot groups [Krantz 1982]: Theorem 2.4. f ∈ 0α (G) if and only if f (gt ) ∈ 3α for every horizontal curve gt . Here 0α (G) and 3α denote, correspondingly, the nonisotropic Lipschitz space of Folland and Stein on R N , and the isotropic Lipschitz space on R. A curve is called horizontal if dtd gt ∈ span{X 1 , . . . , X m }. 3. Variational problems We apply the concentration-compactness principle of Lions [1985a; 1985b] in the homogeneous setting of a Carnot group G, to prove that, for any 1 < p < Q, the best constant in the Folland–Stein embedding (1-1) is achieved. A simple argument shows that, without any loss of generality, we can consider only nonnegative functions; we shall do that throughout this section. Consequently, we show that, for any such p, the quasilinear equation with critical exponent, (3-1)

Lpu =

n X

∗ X j |X u| p−2 X j u = −u p −1

in G,

j=1

possesses a weak nonnegative solution that is also, up to a constant, an extremal for the following variational problem Z Z def p ∞ p∗ |u| = 1 . (3-2) I = inf |X u| u ∈ C0 (G), G

G

We used that L p (cu) = c p−1 L p u to reduce the equation given by the Euler– Lagrange multiplier to (3-1). We shall consider a similar problem when restricting the test functions to those having with a certain symmetry. We prove that the corresponding infimum is achieved again. The purpose of this section is to record such basic results. It is still an open question to find the norm of the Folland–Stein embedding. An interesting and more accessible problem is obtained by requiring that the group be of Heisenberg type. The precise value of the norm of the embedding when considering only functions with symmetries and p = 2 was found in [Garofalo and Vassilev 2001]. The proof required the existence result of this section. In

368

DIMITER VASSILEV

the particular case of the Heisenberg group, this problem was solved without the symmetry restriction by Jerison and Lee [1988]. 3A. The best constant in the Folland–Stein inequality. A minimizing sequence {u n } ∈ C0∞ (G) of the variational problem (3-2) is characterized by the properties Z Z p∗ (3-3) |u n | = 1 and |X u n | p −−−→ I. G

n→∞

G

Two crucial aspects of equation (3-1) and of the above variational problem are their invariance with respect to group translations and dilations. The former is obvious, since the vector fields X j are left-invariant. The latter must be suitably interpreted, and follows from the observation that L p (u ◦ δλ ) = λ p δλ ◦ L p u.

(3-4)

If we thus define, for a solution u of (3-1) and for λ > 0, the rescaled function u λ = λα u ◦ δλ , then it is clear that u λ satisfies (3-1) if and only if α = Q/ p ∗ = (Q − p)/ p. These considerations lead to the introduction, for u ∈ C0∞ (G), of two new functions: def

τh u = u ◦ τh ,

(3-5)

h ∈ G,

where τh : G → G is the left-translation operator τh (g) = hg; and def

∗

u λ = λ Q/ p u ◦ δλ ,

(3-6)

λ > 0.

It is easy to see that the norms in the Folland–Stein inequality and the functionals in the variational problem (3-2) are invariant under the transformations (3-5) and (3-6). Only the second part requires a small computation, since d H is bi-invariant under translations. Z Z ∗ ∗ p∗ p∗ kδλ uk L p∗ (G) = |u(δλ g)| p d H (g) = |u(g)| p λ−Q d H (g) = λ−Q kuk L p∗ (G) . G

G

This shows that ku λ k L p∗ (G) = kuk L p∗ (G) . Similarly, p

kδλ uk ˚ 1, p D

p

(G)

p

p

= kλδλ X uk L p∗ (G) = λ p kδλ X uk L p∗ (G) = λ p−Q kuk ˚ 1, p D

(G)

.

Taking into account that p ∗ = p Q/(Q− p), we obtain kX u λ kD˚ 1, p (G) = kX ukD˚ 1, p (G) . The main result about the existence of global minimizers is this: Theorem 3.1. Let G be a Carnot group, and consider the minimization problem ˚ 1, p (G), (3-2). Every minimizing sequence {u n } of (3-2) is relatively compact in D possibly after translating and dilating each of its elements using (3-5) and (3-6). In particular, there exists a minimum of (3-2), and the equation (3-7)

L p u = −u p

∗ −1


369

˚ 1, p (G). admits a nontrivial nonnegative solution u ∈ D The proof of Theorem 3.1 is based on an adaptation of the concentration-compactness principle. In such an adaptation, the Euclidean space Rn is replaced by a Carnot group G, with its homogeneous structure and Carnot–Carathéodory distance. We mention that the implementation of Lions’ program relies, among others, on the Rellich–Kondrachov compact embedding. In the subelliptic setting, the proof of this result requires a substantial amount of work. A general version of it was proved in [Garofalo and Nhieu 1996]; it states that, if denotes a bounded X -PS domain (Poincaré–Sobolev domain) in a Carnot–Carathéodory space, then the embedding L1, p () ⊂ L q () is compact provided that 1 ≤ q < p ∗ = p Q/(Q − p). Here, L1, p () indicates the Sobolev space of those functions f ∈ L p () such that X f ∈ L p (), endowed with the natural norm. Carnot groups are the basic models of Carnot–Carathéodory spaces. We shall apply such a result to an increasing sequence of bounded domains k ⊂ k+1 ⊂ G such that k % G. As k , we can take the Carnot–Carathéodory ball Bk centered at the identity e ∈ G with radius k, since it was proved in [Franchi et al. 1994; Garofalo and Nhieu 1996] that such sets are X -PS domains. An important tool is the concentration function of a measure: Definition 3.2. For a nonnegative measure dν on G, define the concentration function Q on [0, ∞) by Z def dν . (3-8) Q(r ) = sup g∈G

Br (g)

For a function f on G, we will call the concentration function of f the concentra∗ tion function of the measure | f | p d H . Similarly to Lions’ works, the crucial ingredients in the solution of the variational problem are the next lemmata. For the proofs, we refer to [Lions 1984a; 1984b]. As already mentioned, we have a suitable version of the Rellich–Kondrachov compact embedding to replace the usual embedding used in the proof of the next lemma: Lemma 3.3. Suppose νn is a sequence of probability measures on G. There exists a subsequence, which we denote by dνn , such that exactly one of the following three conditions holds: (1) Compactness: There is a sequence (gn ) ∈ G such that, for every ε > 0, there exists R > 0 so that, for every n, Z dνn ≥ 1 − ε. B(gn ,R)

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DIMITER VASSILEV

(2) Vanishing: For all R > 0, we have Z lim sup n→∞

g∈G

dνn = 0. B(g,R)

(3) Dichotomy: There exists a λ with 0 < λ < 1, such that, for every ε > 0, there exist R > 0 and a sequence (gn ) with the property: Given R 0 > R, there exist nonnegative measures νn1 and νn2 for which 0 ≤ νn1 + νn2 ≤ νn , supp νn1 ⊂ B(gn , R) and supp νn2 ⊂ G \ B(gn , R 0 ), Z Z λ − νn1 + (1 − λ) − νn2 ≤ ε. Remark 3.4. Using diagonal subsequences, we can also achieve supp νn1 ⊂ B Rn (gn ) and supp νn2 ⊂ G \ B2Rn (gn ), Z Z 1 2 lim λ − νn + (1 − λ) − νn = 0.

n→∞

G

G

˚ 1, p (G), µn = |X u n | p d H + µ, and Lemma 3.5. Suppose u n + u weak-∗ in D ∗ p νn = |u n | d H + ν weak-∗ in measure, where µ and ν are bounded nonnegative measures on G. There exist points gj ∈ G and real numbers dj ≥ 0 and ej ≥ 0, at most countably many nonzero, such that ∗

ν = |u| p +

P

dj δgj

j

µ ≥ |X u| p d H +

P

ej δgj

j ∗

I dj p/ p ≤ ej , where I is the constant from (3-2). In particular, P

∗

dj p/ p < ∞.

˚ 1, p (G) Proof of Theorem 3.1. Since p > 1, we can assume that u n + u weak-∗ in D 0 ˚ 1, p (G), by regarding D ˚ 1, p (G) as the dual of D1, p (G), where p 0 is for some u ∈ D the Hölder conjugate of p. From the Folland–Stein embedding theorem, this is also ∗ true for the weak-∗ convergence in L p (G). We can also assume the minimizing sequence is a.e. pointwise convergent on G. This follows easily from Rellich’s theorem, applied successively to an exhaustion of G by an increasing sequence of Carnot–Carathéodory balls.


371

The sequential lower semicontinuity of the norms shows that kuk L p∗ (G) ≤ limku n k L p∗ (G) = 1, kukD˚ 1, p (G) ≤ limku n kD˚ 1, p (G) = I 1/ p . R ∗ Thus, it is enough to prove that G |u| p = 1, since, by the previous relations and p the Folland–Stein inequality, and noticing that I = 1/S p , we have Z (3-9)

p

I≥

Z

|X u| ≥ I

|u|

G

p∗

p/ p∗

Z =I

∗

|u| p = 1,

when G

G

which would give that u is a minimizer. In other words, we reduce the proof to ∗ showing that u n → u in L p (G), as weak-∗ and norm convergence imply strong convergence. Because of translation and dilation invariance, all the mentioned properties hold if we replace (u n ) with any translated and rescaled sequence, which we shall denote by (vn ). We will consider the following measures, def

∗

dνn = |vn | p d H,

def

dµn = |X vn | p d H.

where vn is a suitable translation and dilation of u n that is to be defined in a moment. Note that dνn is also a sequence of probability measures. From the weak-∗ compactness of the unit ball, without loss of generality we can assume that dνn + dν and dµn + dµ in the weak-∗ topology of all bounded nonnegative measures. R ∗ p The desired convergence, that is, the fact that G |v| = 1, will be obtained by applying the concentration-compactness principle, exactly as in [Lions 1985a] (see also [Struwe 1990]). We shall see that ν is a probability measure, as well as that R ∗ dν = G |v| p . Here v is the limit of the sequence (vn ) taken in various spaces, as we did for the sequence (u n ). Let Qˆ n (r ) be the concentration function of u n , that is, Z def p∗ ˆ (3-10) Q n (r ) = sup |u n | d H . h∈G

Br (h)

Clearly, Qˆ n (0) = 0, limr →∞ Qˆ n (r ) = 1, and Qˆ n is a continuous nondecreasing function. Therefore, for every n we can find an rn > 0 such that 1 1 = . (3-11) Qˆ n rn 2 Since the integral in (3-10) is absolutely continuous, it defines a continuous func∗ tion of h, which, as u n ∈ L p (G), tends to zero when d(h, e) → ∞. Consequently,

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the supremum is achieved, that is, for every n there exist a h n ∈ G such that Z 1 ˆ = |u n (g)| d H (g). (3-12) Qn rn B1/rn (h n ) The concentration functions Q n of the dilated and translated sequence def

∗

vn = rn−Q/ p δrn−1 τgn u

(3-13) satisfy (3-14)

Q n (1) =

Z B1 (e)

1 Q n (1) = , 2

and

dνn

where we have set gn = δdn h −1 n . The homogeneity properties of the metric are essential for proving (3-14). From the definition of vn and (2-8), Z Z ∗ ∗ |vn (h)| p d H (h) = rn−Q |u n (δrn−1 τgn h)| p d H (h) (3-15) Br (g) {d(g,h) 1 and 0 ≤ ϕ ≤ 1, it follows that 1 ≥ ϕn p + (1 − ϕn ) p , and thus εn ≥ −C

(3-31)

Z

|vn | p |X ϕn | p d H − ε

Z

An

|X vn | p d H. G

First we use |X ϕn | ≤ C/Rn , and then we apply Hölder’s inequality on An , ∗

Rn−1 kvn k L p (An ) ≤ Rn−1 |An |1/ p − 1/ p kvn k L p∗ (A ) . n

Since 1/ p − 1/ p ∗ = 1/Q and, from the paragraph above (2-8), |An | ∼ RnQ ,

(3-32) we obtain

Rn−1 kvn k L p (An ) ≤ Ckvn k L p∗ (An ) .

(3-33)

The last term in the above inequality can be estimated as follows: (3-34)

p∗ kvn k L p∗ (A ) n

Z

Z

Z

dνn = dνn − dνn G G\An Z Z Z 1 ≤ dνn − dνn − dνn2 G G\An G\An Z Z Z 1 = dνn − dνn − dνn2 . =

An

G

G

G

Hence, the claim (3-30) follows from (3-35)

Rn−1 kvn k L p (An )

Z ≤C G

1/ p∗ Z Z 1 2 dνn − dνn − dνn →0 G

G

as n → ∞.

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DIMITER VASSILEV

Continuing with the proof that the dichotomy case does not occur, we use the definition of I , see (2-1) and (3-2), together with the above inequalities, to get p

kvn k ˚ 1, p D

p

(G)

= kϕm vn k ˚ 1, p

p

+ εn p p ≥ I kϕm vn k L p∗ (G) + k(1 − ϕn )vn k L p∗ (G) + εn Z p/ p∗ Z p/ p∗ ≥I dνn + dνn + εn D

(G)

+ k(1 − ϕn )vn k ˚ 1, p D

B Rn (gn )

Z ≥I G

dνn1

(G)

G\B Rn (gn )

p/ p∗

Z + G

dνn2

p/ p∗

+ εn .

Letting n → ∞, we obtain p

I = lim kvn k ˚ 1, p n→∞

D

(G)

∗ ∗ ≥ I λ p/ p + (1−λ) p/ p − ε I,

which is a contradiction with the choice of ε in (3-27), and hence the dichotomy case of Lemma 3.3 cannot occur. The proof of the theorem is finished. 3B. The best constant in the presence of symmetries. We consider here the same problem as before, but we restrict the class of test functions. Definition 3.6. Let G be a Carnot group with Lie algebra g = V1 ⊕ V2 · · · ⊕ Vn . We say that a function U : G → R has partial symmetry with respect to g0 if there exist an element g0 ∈ G such that for every g = exp(ξ1 +ξ2 +· · ·+ξn ) ∈ G one has U (g0 g) = u |ξ1 (g)|, . . . , |ξn−1 (g)|, ξn (g) , for some function u : [0, ∞) × · · · × [0, ∞) × Vn → R. A function U is said to have cylindrical symmetry if there exist g0 ∈ G and ϕ : [0, ∞) × · · · × [0, ∞) → R for which U (g0 g) = ϕ |ξ1 (g)|, |ξ2 (g)|, . . . , |ξn (g)| , for every g ∈ G. p ˚ 1, ˚ 1, p We also define the spaces D ps (G) and Dcyl (G) by def p ˚ 1, ˚ 1, p () u(g) = u |ξ1 (g)|, . . . , |ξn−1 (g)|, ξn (g) , (3-36) D ps (G) = u ∈ D ˚ 1, p (G) def ˚ 1, p () u(g) = u |ξ1 (g)|, |ξ2 (g)|, . . . , |ξn (g)| . (3-37) D = u∈D cyl

The effect of the symmetries (see also [Lions 1985b]) is manifested in the fact that, if the limit measure given by Lemma 3.5 concentrates at a point, then it must concentrate on the whole orbit of the group of symmetries. Therefore, in the cylindrical case there could be no points of concentration except at the origin,


377

while in the partially symmetric case the points of concentration lie in the center of the group. p p∗ ˚ 1, Theorem 3.7. (1) The norm of the embedding D ps (G) ⊂ L (G) is achieved. ˚ 1, p (G) ⊂ L p∗ (G) is achieved. (2) The norm of the embedding D cyl

To rule out the dichotomy case in the first part of the theorem, we prove: Lemma 3.8. Under the conditions of Lemma 3.3, the points gn in the dichotomy part can be taken from the center of the group. Proof. Define the concentration function of νn by Z ps def (3-38) Q n = sup dνn . h∈C(G)

Br (h)

The rest of the proof is identical to the proof of Lemma 3.3, with the remark that, ps in the dichotomy part, the definition of Q n shows that the points gn can be taken to belong to the center. Proof of Theorem 3.7. We argue as for Theorem 3.1. p ˚ 1, ˚ 1, p () leads to the variational (1) Finding the norm of the embedding D ps (G) ⊂ D problem (3-39)

ps def

I ps ≡ I1 = inf

Z G

p ˚ 1, |X u| p : u ∈ D ps (G),

Z

∗ |u| p = 1 . G

We take a minimizing sequence (u n ), that is, Z Z ∗ (3-40) |u n | p = 1 and |X u n | p −−−→ I ps . G

G

n→∞

1, p

˚ ps (G) is invariant under the dilations (3-6). Using the Baker– It is clear that D p ˚ 1, Campbell–Hausdorff formula, it is easy to see that D ps (G) is also invariant under the translations (3-5) by elements in the center C(G) of G. In order to extract a suitable dilated and translated subsequence of {u n }, we have to make sure that we translate always by elements belonging to C(G). For this, we define the concentration function of u n as Z def ps p∗ ˆ (3-41) Q n (r ) = sup |u n | d H . h∈C(G)

Br (h)

We can fix rn > 0 and h n ∈ C(G) such that (3-11) and (3-12) hold. Define the p ˚ 1, sequence {vn } as in (3-13). As mentioned above, vn ∈ D ps (G) as well. Equation (3-15) holds without any changes. By taking the supremum over g ∈ C(G) we obtain (3-16), and by using (3-17) we obtain (3-14). At this point, we apply Lemma 3.8. The case of vanishing is ruled out from the normalization (3-14) of

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DIMITER VASSILEV

the sequence {vn }. Suppose we have dichotomy. As before, we take a sequence Rn > 0 such that (3-25) and (3-26) hold. We choose a cut-off function ϕ from the p ˚ 1, space D ps (G), also satisfying supp ϕ ⊂ 2 (e)

(3-42)

ϕ≡1

and

on 1 (e),

where r (g) denotes a gauge ball centered at g and of radius r , that is, r (g) = {h ∈ G | N (h −1 g) < r }.

(3-43)

This can be done by setting ϕ = η(N (g)), where η(t) is a smooth function on the real line, supported where |t| < 2 and with η ≡ 1 on |t| ≤ 1. We define the cut-off functions ϕn as before, by ϕn = δ Rn−1 τgn ϕ. From Baker–Campbell–Hausdorff, they have partial symmetry with respect to the identity, since gn ∈ C(G) (see Lemma 3.8), and the gauge is a function with partial symmetry G. By setting An = 2Rn (gn ) \ Rn (gn )

(3-44) and noting that

|An | ∼ RnQ ,

(3-45)

we see that (3-30) holds. Now, from the definition of Ips , keeping in mind that ϕn and vn have partial symmetry with respect to the identity, we obtain p

kvn k ˚ 1, p D

p

(G)

p + k(1 − ϕn )vn k ˚ 1, p + εn (G) D (G) p p Ips kϕm vn k L p∗ (G) + k(1 − ϕn )vn k L p∗ (G) + εn Z p/ p∗ Z p/ p∗

= kϕm vn k ˚ 1, p D

≥

≥ Ips

dνn

dνn

+

B Rn (gn )

Z ≥ Ips ≥ Ips λ

G p/ p ∗

dνn1

G\B Rn (gn )

p/ p∗

Z + G

dνn2

p/ p∗

+ εn

+ εn

∗ + (1−λ) p/ p + εn .

Letting n → ∞, we come, as in (3-30), to (3-46)

∗

p

lim kvn k ˚ 1, p

n→∞

D

(G)

≥ Ips λ p/ p + (1−λ) p/ p

∗

p

> Ips ,

since 0 < λ < 1 and p/ p ∗ < 1. This contradicts that kvn k ˚ 1, p → I ps as n → ∞, D (G) which shows that the dichotomy case of Lemma 3.8 cannot occur. Hence, the compactness case holds. As in Theorem 3.1, we see that Z dν = 1. G


379

Next, we apply Lemma 3.5, with I replaced by Ips . The important fact here is that the partial symmetry of the sequence {vn } implies that the points of concentration of dν, if they occur, must be in the center of the group. Having this in mind together with the definitions of the concentration functions, we can justify the validity of (3-24), and finish the proof of part (1). (2) The vanishing case is ruled out by using the dilation (but not translation, because of the symmetries) invariance, and by normalizing the minimizing sequence with the condition Q n (1) = 1/2 ; see (3-14). Suppose that the dichotomy case occurs. We shall see that this leads to a contradiction. The points {gn } in the dichotomy part of Lemma 3.3 must be a bounded sequence. If not, let ε = λ/2 and R as in the lemma. Because of the invariance under rotations in the layers of the functions vn and the Haar measure d H (which is just the Lebesgue measure), for any arbitrarily fixed natural number N0 we can find a point gn and N0 points on the orbit of gn under rotations in one of the layers, such that the balls with radius R centered at all these points do not intersect. This leads to a contradiction, since the integral of the probability measure dνn over each of these balls is greater than λ/2. Thus, {gn } is a bounded sequence. This is, however, impossible since dνn are probability measures. Therefore, the compactness case holds. As in the dichotomy part, we see that the sequence {gn } dν can concentrate only at the origin e. This is impossible. 4. Global boundedness of weak solutions Let p ∈ (1, Q) and denote by p ∗ the Sobolev conjugate p ∗ = p Q/(Q − p), and by ˚ 1, p () be a weak solution, not p 0 the Hölder conjugate p 0 = p/( p − 1). Let u ∈ D necessarily bounded, of the equation (1-3) in an open set ⊂ G. “Weak solution” means that, for every ϕ ∈ C0∞ (), we have Z (4-1)

|X u|

p−2

hX u, X ϕi d H =

Z

|u| p

∗ −2

u ϕ d H.

∗ ∗ ∗ ∗ 0 ˚ 1, p (), we Note that u p −1 ∈ L p /( p −1) () = L ( p ) . From the definition of D ˚ 1, p (). The main result of this section is obtain that (4-1) holds for every ϕ ∈ D that weak solutions as above are bounded functions. In the next theorem we prove a more general result.

˚ 1, p () be a weak solution to the equation Theorem 4.1. Let u ∈ D (4-2)

m X i=1

X i (|X u| p−2 X i u) = −V |u| p−2 u

in ,

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DIMITER VASSILEV

that is, Z (4-3)

|X u|

p−2

hX u, X ϕi d H =

Z

V |u| p−2 u ϕ d H,

for every ϕ ∈ C0∞ (). (1) If V ∈ L Q/ p (), then u ∈ L q () for every p ∗ ≤ q < ∞. (2) If V ∈ L t () ∩ L Q/ p () for some t > Q/ p, then u ∈ L ∞ (). Proof. The assumption that V ∈ L Q/ p (), together with the Folland–Stein in˚ 1, p (). This can be seen by equality, shows that (4-3) holds true for any ϕ ∈ D ˚ 1, p () by a sequence of test functions ϕn ∈ C ∞ (), approximating in the space D 0 which will allow us to put the limit function in the left-hand side of (4-3). On ∗ the other hand, the Folland–Stein inequality implies that ϕn → ϕ in L p (). Set t0 = Q/ p, and its Hölder conjugate t00 = t0 /(t0 − 1). An easy computation gives 1 p−1 1 1 + = 1− ∗ = ∗ 0. ∗ t0 p p (p ) Hölder’s inequality then shows that V |u| p−2 u ∈ L ( p ) (), which allows us to pass to the limit in the right-hand side of (4-3). We turn to the proofs of (1) and (2). ˚ 1, p () ∩ L q () with q ≥ p ∗ , then u ∈ L κq (1) It is enough to prove that, if u ∈ D ∗ 0

with κ = p ∗ / p > 1. Let G(t) be a continuous, piecewise-smooth, and globally Lipschitz function on the real line, and set Z u (4-4) F(u) = |G 0 (t)| p dt. 0

Clearly, except at finitely many points, F is a differentiable function with a bounded and continuous derivative. From the chain rule (see [Garofalo and Nhieu 1996], for ˚ 1, p () is a legitimate test function in (4-3). example) there follows that F(u) ∈ D The left-hand side, taking into account that F 0 (u) = |G 0 (u)| p , can be rewritten as Z Z |X u| p−2 hX u, X F(u)i d H = |X G(u)| p .

The Folland–Stein inequality (1-1) gives Z p/ p∗ Z p−2 p∗ (4-5) |X u| hX u, X F(u)i d H ≥ S p |G(u)| .

We choose G(t) as G(t) =

 (sign t)|t|q/ p

if 0 ≤ |t| ≤ l,

l (q/ p) −1 t

if l < |t|.


381

From the power growth of G, besides the preceding properties, this function also satisfies (4-6)

|u| p−1 |F(u)| ≤ C(q)|G(u)| p ≤ C(q)|u|q .

The constant C(q) depends also on p, but for us this is a fixed quantity. At this moment, the value of C(q) is not important, but an easy calculation shows that C(q) ≤ Cq p−1 , with C depending on p; we will use this in part (2). Note that p t00 = p ∗ . Leave M > 0 to be fixed in a moment, and estimate the integral in the right-hand side of (4-3): Z (4-7) V |u| p−2 u F(u) d H Z Z = V |u| p−2 u F(u) d H + V |u| p−2 u F(u) d H (|V |≤M)

(|V |>M)

Z

|u| p−1 F(u) d H

≤M (|V |≤M)

Z

t0

1/t0 Z

|V | d H

+

|u|

(|V |>M)

Z

p−1

t 0 F(u) 0 d H

1/t00

|G(u)| p d H

≤ C(q) M

Z + C(q)

t0

1/t0 Z

|V | d H (|V |>M)

p/ p∗ |G(u)| d H . p∗

At this point, we fix an M sufficiently large so that Z 1/t0 Sp t0 C(q) |V | d H ≤ , 2 (|V |>M) which can be done because V ∈ L t0 . Putting together (4-5) and (4-7) we come to our main inequality, Z p/ p∗ Z Z Sp p∗ p ≤ C(q) M |G(u)| d H ≤ C(q) M |u|q d H. |G(u)| d H 2 By the Fatou and Lebesgue dominated convergence theorem, we can let l in the definition of G go to infinity, and obtain Z p/ p∗ Z Sp qp ∗ / p |u| dH ≤ C(q) M |u|q d H. 2 The proof of (1) is finished.

382

DIMITER VASSILEV

(2) It is enough to prove that the L q () norms of u are uniformly bounded by some sufficiently large but fixed L q0 norm of u, q0 ≥ p ∗ , which is finite from (1). We shall do this by iteration. We use the function F(u) from part (1) in the weak form (4-3) of our equation. The left-hand side is estimated from below as before, in (4-5). This time, though, we use Hölder’s inequality to estimate from above the right-hand side: Z

(4-8) V |u| p−2 u F(u) d H ≤ kV kt |u| p−1 F(u) t 0

q ≤ kV kt C(q)|G(u)| p t 0 ≤ C(q)kV kt kukqt 0 . With the estimate from below, we come to p

q

S p kG(u)k p∗ ≤ C(q)kV kt kukqt 0 . Letting l → ∞, we obtain

q/ p p

|u| ∗ ≤ C(q) kV kt kukq 0 . qt p Sp

(4-9)

Set δ = p ∗ /( pt 0 ). The assumption t > Q/ p implies δ > 1, since the latter is equivalent to t 0 < p ∗ / p = t00 , as t0 = Q/ p. With this notation we can rewrite (4-9) as C(q) 1/q 1/q (4-10) kukδqt 0 ≤ kV kt kukqt 0 . Sp Recall that C(q) ≤ Cq p−1 . At this point, we define q0 = p ∗ t 0 and qk = δ k q0 . After a simple induction, we obtain (4-11)

kukqk ≤

k−1 Y

p−1 t 0 /qj

Cqj

t0

Pk−1

kV kt

j=0

1/qj

kukq0 .

j=0

We observe that the right-hand side is finite, (4-12)

∞ ∞ X 1 1 X 1 = 1. Letting j → ∞, we obtain kuk∞ ≤ Ckukq0 .

Remark 4.2. When is a bounded open set, we trivially have V ∈ L Q/ p () whenever V ∈ L t () with t > Q/ p. Also, in this case one can obtain a uniform ∗ estimate of the L ∞ () norm of u by its L p () norm that does not depend on the distribution function of V , as we had in the preceding theorem. This can even be


383

achieved in the unbounded case, assuming only that V ∈ L Q/ p (), but requiring u ∈ L p (). With the previous theorem proved, we turn to our original equation (1-3): ˚ 1, p () is a Theorem 4.3. Take p ∈ (1, Q) and let ⊂ G be an open set. If u ∈ D weak solution to the equation Lpu =

n X

∗ X j |X u| p−2 X j u = −|u| p −2 u

in ,

j=1

then u ∈ L ∞ (). ∗

Proof. We define V = |u| p − p . From the Folland–Stein inequality we have u ∈ ∗ ∗ ∗ L p (), and thus V ∈ L p /( p − p) (). Since p ∗ /( p ∗ − p) = Q/ p, part (1) of ∗ Theorem 4.1 shows that u ∈ L q () for p ∗ ≤ q < ∞. Therefore, V ∈ L q/( p − p) () for any such q and thus, by part (2) of the same theorem, we conclude that u ∈ L ∞ (). 5. Regularity near the characteristic boundary We start by introducing the geometric assumptions on the domain, and describe the regularity of the weak solutions to the Yamabe equation (1-4), which can be obtained from well-known results. Let ⊂ G be a C ∞ domain whose boundary ∂ is an orientable hypersurface. We assume the existence of ρ ∈ C ∞ (G) and γ > 0 such that, for some R ∈ R, (5-1)

= {g ∈ G | ρ(g) < R},

and such that |D ρ(g)| ≥ γ > 0 for every g in some relatively compact neighborhood K of ∂. We shall denote by U a bounded open set such that ∂ ⊂ U , and set ω = ∩ U . We stress that, with this assumption, ∂ω is a compact set inside . However, in Sections 5B, 5C, and 5E we shall make a different assumption, requiring that U contain only the characteristic points of the boundary, and hence ∂ω will reach ∂. The assumptions on the domains are: A-condition: There exist A, r0 > 0 such that, for every g ∈ ∂ and r ∈ (0, r0 ), (G \ ) ∩ B(g, r ) ≥ A |B(g, r )|. (5-2) •

Convexity: There exists M1 > 0 such that the defining function ρ of satisfies the differential inequality •

(5-3)

Lρ ≥

2 hXρ, X ψi M1

in ω,

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DIMITER VASSILEV def

where ψ(g) = |x(g)|2 . We remark that the A-condition is fulfilled if, for example, satisfies the uniform corkscrew condition, see [Capogna and Garofalo 1998; Capogna et al. 1998]. These papers contain an extensive study of examples of domains that, in particular, satisfy (5-2). The A-condition allows us to adapt to the present setting Moser’s [1961] classical iteration arguments. On the other hand, the convexity condition allows the construction of useful barriers. It is satisfied if, for example, ρ is a strictly L-superharmonic function. From Theorem 4.3, we know that u ∈ L ∞ (). This crucial information allows us to implement the local regularity theory of Folland and Stein [1974; Folland 1975] to conclude that u ∈ C ∞ (). The A-condition (5-2) permits us to adapt to the present setting the classical arguments that lead, via Moser’s iteration, to u ∈ 0 0,α () for some 0 < α < 1; see for example [Gilbarg and Trudinger 1983, Section 8.10]. Extending u by zero outside , we can assume henceforth that (5-4)

u ∈ 0 0,α (G) ∩ C ∞ ().

Given the domain and an open neighborhood U , fixed as in the preceding discussion, we assume that M0 > 0 has been chosen so that it fulfill the condition U ⊂ g ∈ G |x(g)|2 ≤ m M0 /4 . (5-5) We shall consider the Riemannian distance d R ( ·, · ) on G, defined using the Euclidean metric on g via the exponential map; that is, if g = exp ξ and h = exp η, we set (5-6)

d R (g, h) = |ξ − η|.

It is straightforward to estimate the Riemannian distance by the Carnot–Carathéodory distance: (5-7)

d R (g, h) ≤ d(g, h).

The estimate in the other direction was proven by Tanaka [1975]. He works in a more general situation than ours, so we state the result as relevant to our setting. Theorem 5.1. Let G be a Carnot group of step r , and take g0 ∈ G. For every ball B(g0 , R) there exists a constant C = C(G, R) such that, if g, h ∈ B(g0 , R), then (5-8)

d(g, h) ≤ C d R (g, h)1/r .

5A. Barrier functions. The barrier functions defined in the next theorem are crucial to the rest of the section. The convexity property of the boundary is essentially what makes these functions useful.


385

Theorem 5.2. Let ⊂ G be a smooth connected bounded open set that satisfies the convexity condition (5-3). Let M ≥ max{M0 , M1 }. For 0 < α ≤ 1, we define 9α = (R − ρ)α e−ψ/M . Under the stated hypothesis, for every g ∈ ω, L 9α (g) ≤ −

(5-9)

m 9 (g). M α

We remark that, when working with L-harmonic functions, one takes α = 1. We also consider α < 1 because of its use in the case of nonlinear equations, for example, Yamabe-type equations. The proof of the above theorem can be found in [Garofalo and Vassilev 2000]. Let η be a smooth vector field defined in U that is transversal to the boundary, that is, d (5-10) η ρ(g0 ) ≡ ρ(gt ) 6= 0, g0 ∈ ∂, t=0 dt where dtd gt = η(gt ). Using the compactness of ∂, we can assume, possibly after taking a smaller U , that there exists a constant δ > 0 such that (5-11)

η ρ(g0 ) ≥ δ > 0

for g0 ∈ ∂.

We note that the transversality condition (5-11) implies that the trajectories of η that start from points of ∂ fill a full open set ω, interior to . Possibly by shrinking the set U , we can assume that ω = ∩ U . To fix the notation, we suppose that there exists t0 ∈ (0, 1) such that gt ∈ ω

for 0 < t < t0 and g0 ∈ ∂.

We shall hereafter use this generic transversal vector field and notation. Lemma 5.3. There exist C1 , C2 > 0 such that, for every g0 ∈ ∂ and 0 ≤ t ≤ t0 , (5-12)

C1 t α ≤ 9α (gt ) ≤ C2 t α .

Proof. Under the assumptions we made, the proof follows from the Taylor formula. Indeed, (5-13) R − ρ(gt ) = tη ρ(g0 ) 1 + O(t) , where O(t) denotes a function bounded by Ct, uniformly in g0 ∈ ∂ and 0 ≤ t ≤ t0 . It is clear from the latter identity that (5-12) follows by using (5-11), the smoothness of ρ, and the boundedness of .

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5B. Lipschitz estimates near the boundary. We start with a theorem about the Green’s function, since its proof is very simple because of the existence of barriers. An important observation is that, thanks to the results of Derridj [1971], the Green’s function is a smooth function up to the noncharacteristic boundary; indeed, its L-Laplacian vanishes near the boundary, and hence it is a smooth function. Therefore, when working with equations with a smooth right-hand side, we shall consider ω = U ∩ , where U is a sufficiently small open neighborhood of the characteristic set 6 defined in (5-37), rather than ∂ ⊂ U as we do in the case of the Yamabe equation; see the definitions of the convexity and A-conditions in Section 5A. Theorem 5.4. Let u ∈ C(ω) ¯ satisfy Lu = 0

in ω,

u=0

on ∂.

If the convexity condition is satisfied in a neighborhood ω of the characteristic boundary 6, then there exists a constant C = C(G, , u) such that (5-14)

|u(g)| ≤ C d R (g, ∂)

for every g ∈ ω.

Remark 5.5. Note that in this theorem the right-hand side uses the Riemannian distance. This is important for the estimates on derivatives that involve vectors not only from the first layer. Clearly, the same inequality holds for the Carnot– Carathéodory distance. Proof. The proof uses 9 as a barrier, and Bony’s maximum principle. As we saw before the statement of the theorem, we have u ∈ C ∞ (ω¯ \ 6). Equation (5-9) with 9 = 91 shows that ±L u(g) = 0 ≥ L C9(g),

g ∈ ω.

On the other hand, for a sufficiently large constant C, we have the estimate (5-15)

C9(g) ≥ |u(g)|

on ∂ω.

To see that (5-15) holds, we argue as follows. Its validity is clear on ∂ since both u and 9 vanish there. On the other hand, the compact set ∂ω ∩ is at a fixed distance away from the characteristic boundary, and the estimate (5-15) follows since 9, u ∈ C ∞ (∂ω ∩ ) and from Lemma 5.3; see the proof of Theorem 5.14 for further details. We can now apply Bony’s maximum principle (Theorem 2.1) and conclude that the bound holds inside as well: |u(g)| ≤ C9(g),

g ∈ ω.


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This implies that u satisfies the Riemannian Lipschitz estimate (5-16)

|u(g)| ≤ Cd R (g, ∂),

g ∈ ω,

which completes the proof.

The next theorem was proved in [Garofalo and Vassilev 2000] by also requiring that be uniformly starlike along 6. A minor modification shows that this is unnecessary. We shall use gt as in (5-10). Theorem 5.6. Let be a C ∞ open bounded set in a Carnot group G, and u a solution of the Yamabe equation (1-4). If the convexity and A-conditions are satisfied, then there exists a constant C = C(G, , u) such that u(g) ≤ C d R (g, ∂)

for every g ∈ ω.

Proof. We begin by observing that, thanks to (5-4) and u = 0 on ∂, we have, for any g0 ∈ ∂, (5-17)

u(gt ) ≤ C d(gt , g0 )α .

Theorem 5.1 now gives, for every g0 ∈ ∂ and 0 ≤ t ≤ t0 , (5-18)

d(gt , g0 ) ≤ Cd R (gt , g0 )1/r ≤ Ct 1/r

for some constant C = C() > 0. Using (5-17) and (5-18), and setting α0 = α/r , we infer that (5-19)

u(gt ) ≤ Ct α0

for every g0 ∈ ∂ and 0 ≤ t ≤ t0 < 1. We now let σ = 2∗ −1 = (Q + 2)/(Q − 2). In the sequel it will be important that σ > 1. Since it is clear that (5-19) continues to hold if in the right-hand side we raise t to any exponent smaller than α0 , we assume in what follows that σ α0 < 1 and that there is some n ∈ N for which (5-20)

σ n α0 = 1.

Next, we use the barriers constructed in Theorem 5.2. For any point gt ∈ ω we have, from (1-4), (5-19), (5-12), and (5-9), that −L u(gt ) = u(gt )σ ≤ Ct σ α0 ≤ CC1−1 9σ α0 (gt ) ≤ −CC1−1 Mm −1 L 9σ α0 (gt ) = −L(C ∗ 9σ α0 )(gt ). Keeping in mind that, as g0 varies in ∂ and t in (0, t0 ), the point gt covers ω, we have proved that L(C ∗ 9σ α0 − u) ≤ 0 in ω.

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At this point, we observe that (possibly using a constant larger than C ∗ ) we also have the estimate (5-21)

C ∗ 9σ α0 ≥ u

on ∂ω.

To see that (5-21) holds, we argue as follows. It is clear that (5-21) holds on ∂ω ∩ ∂, since both u and 9σ α0 vanish there. On the other hand, the compact set ∂ω ∩ is at a fixed distance away from the boundary, and therefore, using that u and 9σ α0 are smooth there and 9σ α0 does not vanish, one trivially obtains the estimate by using the maximum of u. We can now apply to ω Bony’s maximum principle from Theorem 2.1 to infer that a similar estimate also holds in ω. From this result and from the right-hand side of (5-12), we conclude that, for every g0 ∈ ω and 0 < t < t0 , u(gt ) ≤ Ct σ α0 ,

(5-22)

which shows that we have improved on (5-19). It is now clear that, by repeating the above arguments n times, where n is as in (5-20), we shall reach the desired conclusion (5-16). 5C. Estimates for the Green’s function. Theorem 5.7. Let u ∈ C ∞ (ω\∂) ¯ ∩ C(ω) ¯ satisfy Lu = 0

in ω,

u=0

on ∂.

If the convexity condition (5-3) is satisfied, then X u ∈ L ∞ (ω). The proof of Theorem 5.7 is an immediate consequence of Theorem 5.4, with the help of the Schauder-type estimates proved in [Danielli and Garofalo 1998]. Proof. Fix an arbitrary g ∈ ω. With r = dist(g, ∂ω)/2, consider the ball B(g, r ) ⊂ ¯ B(g, r ) ⊂ ω. Applying the interior Schauder estimates to the L-harmonic function u, one has (5-23)

|X u(g)| ≤

C sup |u|, r B(g,r )

for some constant C = C(G). At this point, we invoke Theorem 5.4, which implies, for any g 0 ∈ B(g, r ), the inequality |u(g 0 )| ≤ C 0 d(g 0 , ∂) ≤ C 0 d(g 0 , g) + d(g, ∂) ≤ 2C 0r, with a constant C 0 = C 0 (G, ω, u). Substitution in (5-23) finishes the proof.


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In the next theorems we show the boundedness of derivatives along some transversal to the boundary vector fields. We denote by 6 the characteristic boundary, which is defined in (5-37). Also, as in Theorem 5.4, ω = U ∩ where U is an open neighborhood of 6. We further set 1 = ∂ ∩ U . Theorem 5.8. Suppose η is a vector field from the center of the Lie algebra of G. Let η˜ be the corresponding left-invariant vector field. Assume further that ⊂ G satisfies the convexity condition (5-3) and η˜ is transversal to 6. If u satisfies Lu = 0

in ω,

u=0

on ∂,

then

η˜ u ∈ L ∞ (ω). 0 < t < t0 , g ∈ ω ∩ ω and Proof. For t very close to 0, we define ω = g exp tη t ∂1t = g0 exp tη g0 ∈ 1 , and consider the difference quotients (5-24)

ϕt =

1 Rexp tη u − u , t

g ∈ ωt ,

where Rh u(g) = u(gh) is the right-translations operator. We claim that there exists a constant C > 0 such that, for all t sufficiently close to 0 and g ∈ ωt , one has |ϕt (g)| ≤ C.

(5-25)

Suppose the claim (5-25) is true. Passing to the limit as t → 0, we conclude that |ηu(g)| ˜ ≤ C for every g ∈ ω, thus establishing the theorem. We turn to the proof of (5-25). Note that L ϕt (g) = 0.

(5-26)

That is, each of the functions ϕt is L-harmonic when u is harmonic. This follows since the considered L-Laplacian is left invariant and, when η is in the center of g, we have that h = exp tη is in the center of the group, and hence the right and left translations by h coincide. From Bony’s maximum principle (Theorem 2.1), it is therefore enough to prove that, for some t1 close to 0, (5-25) holds for g ∈ ∂ωt and t ∈ (0, t1 ). Note that ∂ωt = 1t ∪ (∂ω ∩ ). We analyze the two portions separately. Since any point g ∈ 1t can be written as g = g0 exp tη for some g0 ∈ 1, we have (5-27)

ϕt (g) =

u(g0 exp tη) − u(g0 ) . t

We recall Theorem 5.4, which gives (5-28)

|u(g0 exp tη)| ≤ C d R (g0 exp tη, g0 ) ≤ Ct,

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with a fixed constant C when g0 belongs to a fixed compact set. This estimate shows that |ϕt (g)| ≤ C,

(5-29)

g ∈ ∂t .

Finally, the same inequality on ∂ω ∩ follows from the C ∞ regularity of u in . In conclusion, we have proved the claim (5-25), and therefore the theorem. Remark 5.9. The same proof can be used to show boundedness of the derivative along the radial vector field, assuming that it is transversal to ∂. The reason is that the corresponding difference quotients are L-harmonic as well. 5D. Regularity of solutions of the Yamabe equation. Let u be a weak nonnegative solution of the Yamabe-type equation (1-4). We note again that, because of the present lack of boundary Schauder estimates (except in the case of the Heisenberg group: see 5.13), in the next theorem the convexity assumption is required to hold globally. Accordingly, for domains in the Heisenberg group, convexity will be assumed only near the characteristic boundary; see Theorem 5.14. Theorem 5.10. Let be a C ∞ open bounded set in a Carnot group G, and u a solution of (1-4). If the convexity assumption (5-3) and the A-condition (5-2) are satisfied, then X u ∈ L ∞ (). Proof. Since u is smooth away from ∂, in order to prove the theorem it will be enough to show that X u ∈ L ∞ (ω),

(5-30)

where ω is fixed as before. ∗ We begin by introducing v = u 2 −1 ∗ Γ , where Γ is the positive fundamental solution of L, that is, L Γ = −δ. According to [Folland 1975, Corollary 2.8], v ∗ ∗ satisfies the equation Lv = −u 2 −1 . Since by (5-4) u 2 −1 is in 0 0,β (G) for some β ∈ (0, 1) (and u is compactly supported in G), from [Folland 1975, Theorem 6.1] we have 2,β

v ∈ 0loc (G).

(5-31) def

Therefore, if we let w = u − v, to prove (5-30) it is enough to show it for w, that is, to prove that X w ∈ L ∞ (ω). We notice that w is L-harmonic, that is, Lw = 0 in . From Theorem 5.6, u(g) ≤ Cd(g, ∂)

for every g ∈ ω.


391

Since we know that u ∈ C ∞ ( \ ω), we conclude that (5-32)

u(g) ≤ Cd(g, ∂)

for every g ∈ ;

see Theorem 2.3. Fix a point g ∈ ω. With r = dist(g, ∂)/2, consider the ball ¯ B(g, r ) ⊂ B(g, r ) ⊂ . Applying the interior Schauder estimates in Theorem 2.2 to the L-harmonic function w − w(g), one has (5-33)

|X w(g)| ≤

C sup (w − w(g)). r B(g,r )

Note that (5-32) gives, for g 0 ∈ B(g, r ), (5-34)

u(g 0 ) ≤ C dist(g 0 , ∂) ≤ C d(g 0 , g) + dist(g, ∂) ≤ Cr.

Since w = u − v, in view of (5-34) and (5-32) one has, for g 0 ∈ B(g, r ), (5-35) w(g 0 ) − w(g) ≤ u(g 0 ) + u(g) + v(g 0 ) − v(g) ≤ C r + |v(g 0 ) − v(g)| . Finally, we observe that (5-31) implies that v ∈ L1,∞ () Applying Theorem 2.3 once more, we conclude that |v(g) − v(g 0 )| ≤ C d(g, g 0 )

for g, g 0 ∈ .

Substitution of this information in (5-35) gives sup (w − w(g)) ≤ Cr.

B(g,r )

Combining the latter inequality with (5-33) brings the sought-for conclusion X w ∈ L ∞ (ω). This finishes the proof of Theorem 5.10. To end this section, we note that one can show the boundedness of the radial derivative of solutions of the Yamabe equation. This was done in [Garofalo and Vassilev 2000, Theorem 4.7], which we state below. The proof requires that G be of step 2 and the considered domain be C ∞ , bounded, connected, uniformly starlike with respect to one of its points, and satisfying (5-2) and (5-3). Theorem 5.11. Let G be a Carnot group of step 2. Consider a C ∞ connected, uniformly starlike, bounded open set ⊂ G satisfying (5-2) and (5-3). If u is a weak solution of (1-4), then (5-36)

Z u ∈ L ∞ ().

5E. The estimates near the characteristic boundary for a domain in the Heisenberg group. Let be a smooth domain in a Carnot group. Denote by 6 its characteristic set with respect to the system X = {X 1 , . . . , X m }, that is, (5-37) 6 = g ∈ ∂ X j (g) ∈ Tg (∂), j = 1, . . . , m .

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Clearly, when the domain is bounded this is a compact subset of the boundary. P In the next well-known theorem we denote by N = rj=1 dim Vj the topological dimension of G. The symbol HN −1 denotes the (N −1)-dimensional Hausdorff measure constructed using the Riemannian distance on G. Theorem 5.12. Let ⊂ G be a C ∞ domain. One has HN −1 (6) = 0. This result is due to Derridj [1972]. A more refined version has been recently proved by Franchi and Wheeden [1997]. We now consider the Heisenberg group Hn and take ⊂ Hn . We shall use the notation used until now, except that U will be a sufficiently small neighborhood of 6 — that is, 6 b U — and ω = U ∩ as before. We also set 1 = ∂ ∩ U . We recall the following Schauder estimates [Jerison 1981a]: Theorem 5.13. Let be a bounded C ∞ domain in the Heisenberg group Hn , and let ϕ ∈ C0∞ (Hn ) be supported in a small neighborhood of a noncharacteristic point g0 ∈ ∂. Given f ∈ 0 k,α (), k ∈ N ∪ {0}, 0 < α < 1, for the unique solution u to the Dirichlet problem for the Kohn sub-Laplacian (5-38)

Lu = f

in ,

u=0

on ∂,

one has ϕu ∈ 0 k+2,α (). Following the arguments in the preceding sections, we can prove the following: Theorem 5.14. Let be a smooth connected bounded open set in Hn , satisfying the A-condition. Suppose that the convexity condition is satisfied in an interior neighborhood ω of its characteristic set. If u is a weak nonnegative solution of the CR Yamabe equation, then the horizontal gradient of u is bounded in . If in addition ∂ is uniformly starlike along 6, then the radial derivative Z u is bounded in . Proof. As already mentioned, u ∈ 0 0,α (Hn ) ∩ C ∞ (). From Theorem 5.13, by 2,α taking into account that (Q + 2)/(Q − 2) > 1, we have u ∈ 0loc ( \ 6). Now we can argue as in Theorem 5.10, which holds as long as we have Theorem 5.6 and Theorem 5.11, with the only difference that now ∂ should be replaced with 1. However, by doing this we see that ∂ω = 1 ∪ (∂ω\1), with ∂ω \ 1 reaching ∂. There are two places where this is important: (1) in Theorem 5.6, for (5-21); (2) in Theorem 5.11, for the bound on ∂ωλ \ ∂λ , in the proof’s last paragraph. We start with (1). The set 3 = ∂ω ∩ \1 is at a fixed distance away from the characteristic set 6. Therefore, for every g0 ∈ 3, there exists j ∈ {1, . . . , m} such


393

that X j ρ(g0 ) 6= 0. By continuity, the trajectories of X j fill a (sufficiently small) full neighborhood Vg0 of g0 . This means that there exists t0 = t (g0 ) > 0 such that every g ∈ ∩ Vg0 can be written as g1 exp t X j for some g1 ∈ ∂ ∩ Vg0 and some t ∈ (0, t0 ). Using the uniform transversality of X j to ∂ in ∩ Vg0 and Taylor’s formula, we infer the existence of C = C(g0 ) > 0 such that R − ρ(g1 exp t X j ) ≥ C|t| (5-39) 2,α ¯ for every g1 ∈ ∂ ∩ Vg0 and 0 < t < t0 . We now use that u ∈ 0loc ( \ 6) to deduce ∗ ∗ the existence of a constant C = C (u, g0 ) > 0 such that

u(g1 exp t X j ) ≤ C ∗ |t| ≤ C ∗ |t|ασ0 for every g1 ∈ ∂ ∩ Vg0 and 0 < t < t0 . The latter inequality and (5-39) allow us to conclude that (5-21) does hold in the set ∩ Vg0 , for a constant depending on u and g0 . By a finite-covering argument, we see that (5-21) continues to hold in the intersection of a small neighborhood of ∂ with 3. We can thus separate from ∂. Once inside , we can use the C ∞ smoothness of u to conclude that (5-21) holds on the remaining portion of ∂ω ∩ as well. This proves (1). The embedding theorem 5.25 in [Folland 1975] implies that (5-40)

2,β

1,β/2

1,β/2

0loc (G) ⊂ 3loc (G) = Cloc (G),

where the latter space denotes the standard Hölder class with respect to the Riemannian distance d R ( ·, · ) on G. The proof of (2) follows from the above embedding. Remark 5.15. The previous theorem can be proved for any group of step 2, by 2,α ¯ requiring also that u ∈ 0loc ( \ 6). With this assumption, we can also prove that ηu ˜ ∈ L ∞ (), provided that η˜ is transversal to 6. Here, η ∈ V2 and η˜ is the corresponding leftinvariant vector field on a group G of step 2. We want to give an example of domains in groups of step 2 satisfying the assumptions of the preceding sections. Let G be a Carnot group of step 2. Using exponential coordinates, we define the function 1/4 (5-41) f ε (g) = (ε2 + |x(g)|2 )2 + 16|y(g)|2 , ε ∈ R. For R > 0 and ε ∈ R with ε2 < R 2 , consider the C ∞ bounded open set (5-42)

R,ε = {g ∈ G | f ε (g) < R}.

When ε = 0, it is clear that R,ε is nothing but a gauge pseudo-ball centered at the group identity e, except that the natural gauge was defined in (2-5) without the

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factor 16. Here, we have introduced such a factor for the purpose of keeping a consistent definition with the case of groups of Heisenberg type. For all practical purposes, the reader can ignore it and identify f 0 in (5-41) with (2-5). For g ∈ G, we set R,ε (g) = {h ∈ G | f ε (g −1 h) < R} = g R,ε . Theorem 5.16. Let G be a Carnot group of step 2. For every ε ∈ R with ε2 < R 2 , the domain R,ε (g) satisfies the A-condition, the convexity condition (globally), and is uniformly starlike. For the proof of this theorem see [Garofalo and Vassilev 2000]. Acknowledgments Apart from small changes and the addition of some updates in the references, this paper is part of the author’s doctoral dissertation at Purdue University, 2000. The author would like to thank his advisor Professor Nicola Garofalo for his guidance, support, and encouragement. The author would also like to acknowledge the careful reading of the anonymous referee and thank him/her for the comments, which improved the reading of the paper. References [Aubin 1976a] T. Aubin, “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire”, J. Math. Pures Appl. (9) 55:3 (1976), 269–296. MR 55 #4288 Zbl 0336.53033 [Aubin 1976b] T. Aubin, “Problèmes isopérimétriques et espaces de Sobolev”, J. Differential Geometry 11:4 (1976), 573–598. MR 56 #6711 Zbl 0371.46011 [Bahri and Coron 1988] A. Bahri and J.-M. Coron, “On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain”, Comm. Pure Appl. Math. 41:3 (1988), 253–294. MR 89c:35053 Zbl 0649.35033 [Bony 1969] J.-M. Bony, “Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés”, Ann. Inst. Fourier (Grenoble) 19:fasc. 1 (1969), 277–304 xii. MR 41 #7486 Zbl 0176.09703 [Brezis 1986] H. Brezis, “Elliptic equations with limiting Sobolev exponents—the impact of topology”, Comm. Pure Appl. Math. 39:S, suppl. (1986), S17–S39. MR 87k:58272 Zbl 0601.35043 [Brézis and Kato 1979] H. Brézis and T. Kato, “Remarks on the Schrödinger operator with singular complex potentials”, J. Math. Pures et Appliquées. (9) 58:2 (1979), 137–151. MR 80i:35135 Zbl 0408.35025 [Brézis and Nirenberg 1983] H. Brézis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents”, Comm. Pure Appl. Math. 36:4 (1983), 437–477. MR 84h:35059 Zbl 0541.35029 [Capogna and Garofalo 1998] L. Capogna and N. Garofalo, “Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics”, J. Fourier Anal. Appl. 4:4-5 (1998), 403–432. MR 2000k:35056 Zbl 0926.35043


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CONTENTS Volume 227, no. 1 and no. 2

Bruce Allison, Stephen Berman and Arturo Pianzola: Iterated loop algebras

1

Stephen Berman with Bruce Allison and Arturo Pianzola

1

Marie Françoise Bidaut-Véron: Self-similar solutions of the p-Laplace heat equation: the fast diffusion case

201

Ildefonso Castro, Haizhong Li and Francisco Urbano: Hamiltonian-minimal Lagrangian submanifolds in complex space forms

43

Eva Maria Feichtner: Complexes of trees and nested set complexes

271

Daniel Goldstein, Robert M. Guralnick, Lance Small and Efim Zelmanov: Inversion invariant additive subgroups of division rings 287 Robert M. Guralnick with Daniel Goldstein, Lance Small and Efim Zelmanov

287

Matthew Horak: A spectral sequence determining the homology of Out(Fn ) in terms of its mapping class subgroups

65

Mikhail G. Katz and Stéphane Sabourau: An optimal systolic inequality for CAT(0) metrics in genus two

95

Edward Kissin and Victor S. Shulman: Operator multipliers

109

Mohammed-Larbi Labbi: Manifolds with positive second Gauss–Bonnet curvature

295

Haizhong Li with Ildefonso Castro and Francisco Urbano M. Manickam and B. Ramakrishnan: An Eichler–Zagier map for Jacobi forms of half-integral weight Arturo Pianzola with Bruce Allison and Stephen Berman

43 143 1

Raphaël Ponge: The tangent grupoid of a Heisenberg manifold

151

B. Ramakrishnan with M. Manickam

143

Stéphane Sabourau with Mikhail G. Katz

95

Victor S. Shulman with Edward Kissin

109

Lance Small with Daniel Goldstein, Robert M. Guralnick and Efim Zelmanov

287

Masao Tsuzuki: Green currents for modular cycles in arithmetic quotients of complex hyperballs

311

Francisco Urbano with Ildefonso Castro and Haizhong Li

43

400

Dimiter Vassilev: Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups

361

Yunyan Yang: Moser–Trudinger trace inequalities on a compact Riemannian surface with boundary

177

Efim Zelmanov with Daniel Goldstein, Robert M. Guralnick and Lance Small

287

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Volume 227

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Self-similar solutions of the p-Laplace heat equation: the fast diffusion case 201 M ARIE F RANÇOISE B IDAUT-V ÉRON Complexes of trees and nested set complexes E VA M ARIA F EICHTNER

271

Inversion invariant additive subgroups of division rings 287 DANIEL G OLDSTEIN , ROBERT M. G URALNICK , L ANCE S MALL AND E FIM Z ELMANOV Manifolds with positive second Gauss–Bonnet curvature M OHAMMED -L ARBI L ABBI

295

311

Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups D IMITER VASSILEV

361

Pacific Pacific Journal Journal of of Mathematics Mathematics

2006

Green currents for modular cycles in arithmetic quotients of complex hyperballs M ASAO T SUZUKI

Pacific Journal of Mathematics

PACIFIC JOURNAL OF MATHEMATICS

Vol. 227, No. 2 Volume 227

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