SOME NEW PDE METHODS FOR WEAK KAM THEORY Lawrence ...

SOME NEW PDE METHODS FOR WEAK KAM THEORY

Lawrence C. Evans 1 Abstract. We discuss a new approximate variational principle for weak KAM theory. The advantage of this approach is that we build both a minimizing measure and a solution of the generalized eikonal equation at the same time. Furthermore the approximations are smooth, and so we can derive some interesting formulas upon differentiating the Euler–Lagrange equation. Our method is inspired by the “calculus of variations in the sup–norm” ideas of Aronsson, Jensen, Barron and others.

1. Introduction. This paper provides a new variational and PDE approach to Mather’s minimization principle in dynamics, set forth in [Mt1-2], and to weak KAM theory, earlier developed in Fathi [F1-3] and in [E-G]. An overall goal of this work is to identify a sort of integrable structure within certain general classes of Hamiltonian dynamics and to understand how dynamical information is encoded in a related “effective Hamiltonian”. See Mather–Forni [M-F], Fathi [F3] or the introduction to [E-G] for more background. We are motivated here by the min–max formula (1.1)

¯ )= H(P

inf

max H(P + Dv, x),

v∈C 1 (Tn ) x∈Tn

where the Hamiltonian H satisfies some conditions listed below, most notably periodicity ¯ denotes the effective Hamiltonian in the sense in x over the unit flat torus Tn . The term H of Lions–Papanicolaou–Varadhan [L-P-V] (which is equivalent in our setting to Mather’s function α). Several authors, among them Fathi, Ma˜ n´e, Contreras–Iturriaga–Paternain and Gomes, have independently derived this identity. For the reader’s convenience, we reproduce in Appendix B (§6) a quick proof that Fathi recently found. A sup–norm variational principle. Formula (1.1) suggests that we can compute ¯ ) by trying to minimize the sup–norm of H(P + Dv, x) over Tn . This viewpoint is H(P 1

Supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley

1

strongly reminiscent of “Aronsson’s variational principle” in the calculus of variations: See for instance Aronsson [A1-2], Barron [B], Barron–Jensen–Wang [B-J-W], etc. The idea is that we should not just try to minimize the sup–norm of H(P + Dv, x) over all of Tn , but rather we should look for a function v which minimizes ||H(P + Dv, x)||L∞ (U ) , relative to its boundary values, for each open subdomain U ⊂ Tn . We then call v an absolute minimizer. The references cited above suggest also that we can search for absolute minimizers by replacing the sup–norm by an Lp -norm and letting p → ∞. That is, for each 1 ≤ p < ∞ we should find v p ∈ C 1 (Tn ) to minimize the functional  Tn

|H(P + Dv p , x)|p dx

and then study the limiting behavior of the functions v p when p → ∞. (Paternain [P] discusses a somewhat related variational formulation, but without sending p → ∞.) It is however more elegant to employ exponentials rather than powers, and some of the resulting computations resemble the formalism of statistical mechanics. We will therefore for this paper first take a positive integer k, and look then for v k ∈ C 1 (Tn ) to minimize the functional  k (1.2) Ik [v] := ekH(P +Dv ,x) dx. Tn

Under the following assumptions on H, there exists a minimizer v k ∈ C ∞ (Tn ), which is unique once we require  v k dx = 0. Tn

See Appendix A in §5 for more details. Hypotheses on the Hamiltonian: We suppose H : Rn × Rn → R, H = H(p, x), is smooth and satisfies these conditions: (i) periodicity in x: For each p ∈ Rn , the mapping x → H(p, x) is Tn -periodic. (ii) strict convexity: There exists a constant γ > 0 such that (1.3)

γ|ξ|2 ≤ Hpi pj (p, x)ξi ξj

for each p, x, ξ ∈ Rn . 2

(iii) growth bounds: There exists a constant C such that |Dp2 H(p, x)| ≤ C (1.4)

2 |Dx,p H(p, x)| ≤ C(1 + |p|)

(p, x ∈ Rn ).

|Dx2 H(p, x)| ≤ C(1 + |p|2 ) These assumptions imply the following growth estimates for H and its first derivatives: (1.5)

(1.6)

γ 2 |p| − C ≤ H(p, x) ≤ C(1 + |p|2 ) 2

(p, x ∈ Rn ),

|Dp H(p, x)| ≤ C(1 + |p|) |Dx H(p, x)| ≤ C(1 + |p|2 )

(p, x ∈ Rn ).

The corresponding Lagrangian is L(q, x) := max(p · q − H(p, x)) p

(q, x ∈ Rn ).

PDE interpretations. The Euler–Lagrange equation for our minimizer of Ik [·] is div(ekH(Du

(1.7)

k

,x)

Dp H(Duk , x)) = 0,

or, equivalently, 1 (Hpi (Duk , x))xi +Hpi (Duk , x)Hpj (Duk , x)ukxi xj k + Hxi (Duk , x)Hpi (Duk , x) = 0, where we have set uk := P · x + v k . Define also k

(1.8)

k ekH(Du ,x) ¯k = ek(H(Du ,x)−H (P )) , σ :=  k ,x) kH(Du e dx Tn

k

for (1.9)

1 H (P ) := log k

Then (1.10)



kH(Duk ,x)

¯k

e Tn

 σ ≥ 0, k

dσ k = 1, Tn

3

 dx .

where dσ k := σ k dx. Observe that the Euler–Lagrange equation (1.7) now reads div(σ k Dp H(Duk , x)) = 0.

(1.11)

Overview. In the following sections we attempt to understand the limiting behavior of uk and σ k as k → ∞. In §2 we extract some convergent subsequences and in part characterize their limits. We then in §3 lift into phase space, as a technical device to help us refine our understanding of the limit function u and the limit measure σ. Section 4 ¯ k , and as an application provides some formulas for the first and second derivatives of H discusses consequences of a nonresonance condition. Appendix A in §5 records a sup–norm estimate on |Duk | and discusses the solvability of (1.7). Appendix B in §6 reproduces Fathi’s short proof of min-max formula (1.1). As we will see, the technique of approximating by the PDE (1.11) is fairly elegant. It is less clear however that this will lead to anything really new, although the structure of (1.11) perhaps suggests that numerical methods for nonlinear elliptic PDE may prove ¯ There are several papers in the PDE literature that study variuseful for computing H. ational problems with exponential growth nonlinearities. These include Duc–Eells [D-E], Lieberman [L] and Naito [N] (which handles Rm -valued mappings). I thank M. Giaquinta and G. Lieberman for these references.

2. Convergence, approximating the effective Hamiltonian. We propose to let k → ∞, and will need some simple estimates. We may hereafter suppose without loss of generality that H ≥ 0. Lemma 2.1. For each 1 ≤ q < ∞, there exists a constant C = C(P, q) such that sup ||uk ||W 1,q (Tn ) ≤ C. k

Proof. Since v k is a minimizer of Ik [·], we can compare Ik [v k ] with Ik [0], to estimate 

kH(P +Dv k ,x)

e Tn

 dx ≤

Tn

ekH(P,x) dx ≤ ekC .

Hence ||eH(Du

k

,x)

||L1 (Tn ) ≤ ||eH(Du 4

k

,x)

||Lk (Tn ) ≤ eC ;

and, since H grows quadratically in variable p, this bound implies for each 1 ≤ q < ∞ that ||Duk ||Lq (Tn ) ≤ C. Since the integral of v k is zero, we can then estimate the Lq -norm of v k , and so to finish the proof.
Utilizing this Lemma and if necessary passing to a subsequence, we have uk → u = P · x + v

uniformly on Tn ,

and for each 1 ≤ q < ∞, weakly in Lq (Tn ; Rn ),

Duk  Du = P + Dv

where v ∈ W 1,q (Tn ). In addition, in view of (1.10) we may also suppose that σk  σ

weakly as measures on Tn ,

where σ is a Radon probability measure on Tn . Theorem 2.1. (i) We have (2.1)

¯ ) = lim H ¯ k (P ) = lim 1 log H(P k→∞ k→∞ k



kH(Duk ,x)

e

 dx .

Tn

(ii) The function u is a viscosity solution of Aronsson’s equation (2.2)

−Hpi (Du, x)Hpj (Du, x)uxi xj = Hxi (Du, x)Hpi (Du, x).

(iii) Furthermore, ¯ ) H(Du, x) ≤ H(P

(2.3)

a.e. in Tn .

Proof. 1. According to Lions, Papanicolaou, and Varadhan [L-P-V], there exists a periodic, Lipschitz continuous function v solving ¯ ) H(P + Dv, x) = H(P

(2.4)

in the viscosity sense, and thus also almost everywhere. Then   ¯ kH(P +Dv k ,x) e dx ≤ ekH(P +Dv,x) dx = ekH(P ) , Tn

Tn

5

and consequently 1 lim sup log k→∞ k



kH(Duk ,x)

e

 dx

Tn

¯ ). ≤ H(P

Suppose next that 1 lim inf log k→∞ k



kH(Duk ,x)

e

 dx

Tn

¯ )−ε < H(P

for some ε > 0. Let Λε,k

 ε n k ¯ . := x ∈ T | H(Du , x) > H(P ) − 2

Then for some sequence kj → ∞, we have 1 ¯ ) − ε. ¯ ) − ε ≤ H(P log |Λε,kj | + H(P kj 2 This inequality implies |Λε,kj | ≤ e−kj 2 . ε

Hence for each measurable set A ⊂ Tn , with |A| > 0,   ¯ ) − ε, − H(Du, x) dx ≤ lim inf − H(Dukj , x) dx ≤ H(P kj →∞ 2 A A the slash through the integral denoting an average, and so ¯ )− H(Du, x) ≤ H(P

ε 2

a.e.

We now set uδ := ηδ ∗ u, where ηδ denotes a standard mollifier with support in the ball B(0, δ). Then ¯ ) − ε in Tn H(Duδ , x) ≤ H(P 4 for sufficiently small δ > 0. This however contradicts the min-max formula (1.1) since uδ = P · x + vδ for some smooth, periodic function vδ . We have proved assertion (i). 2. Let u − φ have a strict maximum at a point x0 ∈ Tn . Then uk − φ has a maximum at a nearby point xk , with xk → x0 as k → ∞. Then (1.7) implies − div(ekH(Dφ,xk ) Dp H(Dφ, xk )) ≤ 0; that is, 1 −(Hpj φxj xi + Hxi )Hpi ≤ (Hpi )xi k 6

at the point xk . Let k → ∞, to deduce −Hpi (Dφ, x0 )Hpj (Dφ, x0 )φxi xj ≤ Hxi (Dφ, x0 )Hpi (Dφ, x0 ). The opposite inequality holds similarly, should u − φ attain a strict minimum at x0 . This proves statement (ii). 3. Let A ⊆ Tn be measurable, with |A| > 0. Then lower semicontinuity and Jensen’s inequality imply   − H(Du, x) dx ≤ lim inf − H(Duk , x) dx k→∞ A A   1 kH(Duk ,x) ¯ ), dx ≤ H(P ≤ lim sup log − e k→∞ k A ¯ ) almost everywhere in Tn . This is (iii). and so H(Du, x) ≤ H(P




3. Minimizing measures. To understand more about the structure of the measure σ, it is convenient to lift into R × Tn , as follows. Define n

µk := δ{q=Dp H(Duk ,x)} σ k ;

(3.1) that is,





 k

Φ(q, x) dµ =

(3.2) Rn

Tn

Tn

Φ(Dp H(Duk , x), x) dσ k

for each C 1 function Φ. n n Lemma 3.1. The family of probability measures {µk }∞ k=1 on R × T is tight.

Proof. 1. To simplify notation, we temporarily drop the various superscripts k. Multiply the Euler–Lagrange PDE div(σDp H) = 0 by ∆u and integrate by parts over Tn , to find  0= (σHpi )xj uxi xj dx Tn  σ(Hpi pl uxl xj + Hpi xj )uxi xj + σkhxj Hpi uxi xj dx, = Tn

for h := H(Du, x). Note that hxj = Hpi uxi xj + Hxj . We substitute into the last term above, and obtain  σHpi pl uxl xj uxi xj + σkhxj hxj dx Tn  −σHpi xj uxi xj + σkhxj Hxj dx. = Tn

7

Observe next that σxj = σkhxj in the last expression, and integrate by parts. After some rewriting, we derive the identity   (3.3) Hpi pl uxl xj uxi xj + khxj hxj dσ = − 2Hpi xj uxi xj + Hxj xj dσ. Tn

Tn

In view of our hypotheses on H, this formula implies the estimate    2 2 k|Dh| dσ ≤ C + C |Du| dσ ≤ C + C h dσ. Tn

2. Then



1 2

Tn

Tn

1 2



|Dσ | + (σ ) dx = 2

Tn

k2 |Dh|2 dσ + 1 ≤ kC + C 4

2

Tn

and so Sobolev’s inequality implies 1   1+θ  θ σ dσ ≤ kC + C Tn

for θ =

2 n−2

kh dσ, Tn

kh dσ

Tn

θ if n ≥ 3, θ any positive number if n = 2. Define γ := 2(1+θ) . Then 1   1+θ   θ ¯ + dσ ≤ kC + C σ dσ ≤ kC + C k(h − H) σ γ dσ Tn

Tn

 ≤ kC + C

Therefore



σ θ dσ

.

Tn

 θ

(3.4)

Tn

1  2(1+θ)

1  1+θ

σ dσ Tn

≤ Ck.

3. Finally, we restore the superscripts k and estimate for a sufficiently large constant m that ¯ k + 1} µk {(x, q) | |q| ≥ m} ≤ σ k {x | h ≥ H  −kθ σ k,θ dσ k ≤ Ce−kθ k 1+θ → 0, ≤e as k → ∞.

Tn




Remark. In Appendix A we derive stronger estimates, showing that in fact the gradients {Duk }∞ k=1 are bounded in the sup–norm, uniformly in k, and hence the measures
{µk }∞ k=1 have uniformly bounded supports. Passing to a subsequence if necessary, we may suppose that µk  µ weakly as measures, and µ is a probability measure on Rn ×Tn , according to the Lemma. Note that σ = projx µ, the projection of µ into Tn . Define also the vector   (3.5) Q := q dµ. Rn

8

Tn

Theorem 3.1. (i) The measure µ is flow invariant; that is, 



(3.6) Rn

Tn

q · Dφ dµ = 0

for all φ ∈ C 1 (Tn ). (ii) The limit  (3.7)

¯ ) H(Duk , x) dσ k = H(P

lim

k→∞

Tn

holds. (iii) The measure µ is a Mather minimizing measure, and in particular 

 ¯ L(q, x) dµ = L(Q),

(3.8) Rn

Tn

¯ ). Q ∈ ∂ H(P

(iv) The function u is differentiable σ-almost everywhere and q = Dp H(Du, x)

(3.9) In particular,

µ-almost everywhere.

 Q := Tn

Dp H(Du, x) dσ.

(v) We have ¯ ) H(Du, x) = H(P

(3.10)

σ-almost everywhere

and div(σDp H) = 0

(3.11)

in Tn .

The PDE (3.10) and (3.11) are, respectively, the generalized eikonal equation and the transport equation, satisfied by u and σ. Note we are only asserting that the PDE (3.10) holds on the support of σ. This is the primary difference between our approach and that of Fathi [F1-3], [E-G]. Recall however that Aronsson’s equation (2.2) is valid throughout Tn , and so it remains an interesting question to understand what, if anything, this PDE has to do with the underlying Hamiltonian dynamics. Proof. 1. Let φ ∈ C 1 (Tn ). Then  Rn

 Tn

 q · Dφ dµ = lim

k→∞

Tn

9

Dp H(Duk , x) · Dφ dσ k = 0,

since div(σ k Dp H) = 0 according to (1.11). This proves (3.6). 2. As a first step towards establishing (3.7), let us show that  ¯ ) ≤ lim inf H(P

(3.12)

H(Duk , x) dσ k .

k→∞

Tn

To confirm this, we compute for each λ > 0 that  k ¯ k (P ) dσ k ¯ (P ) = H H n T  k k ¯ (P ) dσ + ¯ k (P ) dσ k H H = ¯ k −λ} ¯ k −λ} {h≥H {h 0. Let ε = 1/2k and sum, to deduce 

∞

Tn

1/2

β1/2k dσ < ∞.

k=1

Hence ∞

(3.18)

1/2

β1/2k (x) < ∞

k=1

for σ-a.e. point x ∈ Tn . Fix such a point x. For some constant ρ > 0, we have  1/2 β2r (x) ≥ ρ −

1/2 |Du − (Du)x,r | dy 2

 ≥ ρ−

B(x,r)

|Du − (Du)x,r | dy,

B(x,r)

where (Du)x,r

 := −

Du dy.

B(x,r)

Thus (3.18) implies

∞ 

− k=1

Since |(Du)x,1/2k+1

B(x,1/2k )

|Du − (Du)x,1/2k | dy < ∞.

 − (Du)x,1/2k | ≤ C −

B(x,1/2k )

|Du − (Du)x,1/2k | dy,

the limit Du(x) := limk→∞ (Du)x,1/2k exists. Next, for each r > 0 we have the estimate  −

 |u(y) − u(x) − Du(x) · (y − x)| dy ≤ Cr −

B(x,r)

|Du(y) − Du(x)| dy = o(r).

B(x,r)

Since u is Lipschitz continuous, in fact max {|u(y) − u(x) − Du(x) · (y − x)|} = o(r) as r → 0,

y∈B(x,r)

and so u is differentiable at x. 6. The foregoing calculations show that Duε (x) → Du(x)

boundedly, σ-almost everywhere. 12

Thus passing to limits in (3.17) gives   |q − Dp H(Du, x)|2 dµ = 0 Rn

Tn

and (3.9) follows. ¯ ) on a set of positive σ measure, we set 7. If H(Du, x) < H(P r = Dp H(Du(x), x) in the inequality in (3.15) and integrate with respect to σ. Calculating as in step 4 above, we deduce that ¯ ¯ ), P · Q < L(Q) + H(P a contradiction. Thus (3.10) is valid, as is (3.11) since    0= q · Dφ dµ = Dp H(Du, x) · Dφ dσ Rn

Tn

Tn

for each φ ∈ C 1 (Tn ).




Remarks. I do not see how to use PDE estimates based upon (1.11) to recover Mather’s theorem that the gradient Du is Lipschitz continuous on the support of the measure σ. (However, see the PDE proof in [E-G].)
¯ k. 4. Some formulas involving H First, we establish a simple monotonicity property. Theorem 4.1. For each P ∈ Rn , we have ¯ k (P ) ≤ H ¯ l (P ) H

(4.1)

if k ≤ l.

Proof. We have  k

Ik [v ] = and so

kH(Duk ,x)

e Tn

 dx ≤

kH(Dul ,x)

e Tn

 dx ≤

lH(Dul ,x)

e Tn

 kl dx

k = Il [v l ] l ;

1

1 Ik [v k ] k ≤ Il [v l ] l .

Take the log of both sides to derive (4.1).




Since the minimization problem (1.2) corresponds to a nonlinearity which is uniformly convex in the variable p, the unique minimizer v k depends smoothly on the parameter ¯ k is a smooth function, and we can compute its first and second P ∈ Rn . Consequently H derivatives: 13

Theorem 4.2. For k = 1, . . . and P ∈ Rn , we have the formulas  k ¯ Dp H(Duk , x) dσ k (4.2) DH (P ) = Tn

and

 2

¯k

D H (P ) = k Tn

(4.3)

2 ¯ k (P )) (Dp H(Duk , x)DxP uk − D H



2 ¯ k (P )) dσ k uk − D H ⊗ (Dp H(Duk , x)DxP

+ Tn

2 2 Dp2 H(Duk , x)DxP uk : DxP uk dσ k .

¯ k is a convex function of P . In particular, H Remark. Our notation means that the (l, m)th component of the first term on the right hand side of (4.3) is  ¯ Pk (P ))(Hp (Duk , x)ukx P − H ¯ Pk (P )) dσ k (Hpi (Duk , x)ukxi Pl − H k j j m m l Tn

and of the second term is

 Hpi pj (Duk , x)ukxi Pl ukxj Pm dσ k .

Tn

Proof. 1. According to (1.9), we have ¯ k (P ) kH

e

 ekH(Du

=

k

,x)

dx.

Tn

Differentiate with respect to Pl : ¯k kH

ke (4.4)

¯k H Pl



=

ekH(Du

k

,x)

kHpi ukPl xi dx

ekH(Du

k

,x)

kHpi (δl,i + vPk l xi ) dx

ekH(Du

k

,x)

kHpl dx.

Tn

 =

Tn

 =

Tn

The last equality holds since uk solves the Euler–Lagrange equation (1.7). We cancel the k, rearrange, and derive formula (4.2). 2. As above, write out the lth component of (4.2), and differentiate with respect to Pm :  k ¯k ¯ k kH k ¯k ¯ (HPl Pm + k HPl HPm ) = ekH(Du ,x) (Hpi ukPm Pl xi + Hpi pj ukPl xi ukPm xj ) dx e Tn  k ekH(Du ,x) Hpi ukPl xi Hpj ukPm xj dx. +k Tn

14

The integral of the first term on the right is zero, according to (1.7), and, recalling (4.2), we can recast the remaining expressions into formula (4.3).
Nonresonance and averaging. As an application of these formulas, we recast and ¯ is differentiable at P and that simplify some ideas from §9 of [E-G]. Assume that H ¯ ) satisfies the nonresonance condition: Q = DH(P Q · m = 0

for each vector m ∈ Zn , m = 0.

¯ k (P ) is bounded as k → ∞. Then Theorem 4.3. Suppose also that D2 H   k k (4.5) lim Φ(DP u (P, x)) dσ = Φ(X) dX k→∞

Tn

Tn

for each continuous, Tn -periodic function Φ. Proof. Observe first that the function k

e2πim·DP u = e2πim·x e2πim·DP v

k

is Tn -periodic. Hence we have    k 2πim·DP uk 0= Dp H(Du , x) · Dx e dσ k Tn  k 2 = 2πi e2πim·DP u m · Dp H(Duk , x)DxP uk dσ k n T k ¯ k (P ) dσ k = 2πi e2πim·DP u m · DP H Tn  k 2 ¯ k (P )) dσ k . + 2πi e2πim·DP u m · (Dp H(Duk , x)DxP uk − DP H Tn

Consequently if m = 0, the identity (4.3) implies   k k 2πim·D u k 2 P ¯ (P )) ¯ k (P ))| dσ k |(m · DP H e dσ | ≤ |m| |Dp H(Duk , x)DxP uk − DP H Tn

Tn −1

≤ |m|(k

1

¯ k (P )) 2 . ∆H

¯ ) = Q and m · Q = 0, we deduce that ¯ k (P ) → DH(P Since DH  k lim e2πim·DP u dσ k = 0, k→∞

Tn





and so

k

lim

k→∞

Tn

k

Φ(DP u (P, x)) dσ = 15

Φ(X) dX Tn

for each periodic function Φ whose Fourier expansion contains only finitely many nonzero terms. Such functions are dense in the sup–norm, and so we deduce (4.5).
Remark. As explained in a different setting in [E-G], Theorem 4.3 provides a partial interpretation of the following heuristics. Suppose that we regard u = u(P, x) as a smooth generating function, inducing the canonical change of variables (p, x) → (P, X), where p = Dx u(P, x), X = DP u(P, x). Then the corresponding Hamiltonian dynamics become  ˙ ¯ X = DH(P) ˙ = 0, P and consequently X(t) = Qt + X0 . According to the nonresonance condition, we then have 1 lim T →∞ T





T

Φ(X(t)) dt =

Φ(X) dX. Tn

0




5. Appendix A: Uniform estimates on Duk . Theorem 5.1. There exists a constant C such that (5.1)

¯ k (P ) + max H(Duk , x) ≤ H n T

C log k k

for k = 2, . . . . ¯ ), this ¯ k (P ) ≤ H(P Remark. Since H grows quadratically in the variable p and H implies the bound (5.2)

max |Duk | ≤ C n T

¯ ) ≤ maxx∈Tn H(Duk , x); for k = 1, . . . . Furthermore, the min–max formula (1.1) implies H(P and hence Theorems 4.1 and 5.1 provide the two–sided bounds (5.3)

¯ k (P ) ≤ H(P ¯ )≤H ¯ k (P ) + C log k H k

for k = 2, . . . . The constant C is bounded for P in bounded subsets of Rn . 16




Proof. 1. To ease notation, we drop the superscripts k. Multiply the Euler–Lagrange equation div(σDp H) = 0 by div(σ p Dp H) for p ≥ 0, and integrate by parts:  (σHpi )xj (σ p Hpj )xi dx 0= Tn

 = Tn

(5.4)

(σ(Hpi )xj + σxj Hpi )(σ p (Hpj )xi + pσ p−1 σxi Hpj ) dx

 = Tn

σ p+1 (Hpi )xj (Hpj )xi + pσ p−1 Hpi σxi Hpj σxj + (p + 1)σ p σxi Hpj (Hpi )xj dx

 A + B + C dx.

=: Tn

Recalling our hypotheses on H, we see that A = σ p+1 (Hpi pm uxm xj + Hpi xj )(Hpj pl uxl xi + Hpj xi ) ≥ σ p+1 (γ 2 |D2 u|2 − C(1 + |Du|)|D2 u| − C(1 + |Du|2 )) ≥ −Cσ p+1 (1 + |Du|2 ). Clearly B ≥ 0, and we further calculate that C = (p + 1)σ p σxi Hpj (Hpi pl uxl xj + Hpi xj )

=

(p + 1) p−1 σ Hpi pl σxi σxl + (p + 1)σ p σxi (Hpj Hpi xj − Hpi pl Hxl ), k

≥

(p + 1) p−1 |Dσ|2 − C(p + 1)σ p |Dσ|(1 + |Du|2 ), γσ k

since σxl = σkhxl = σk(Hpj uxl xj + Hxl ). 2. We combine the foregoing estimates, to compute   p+1 p−1 2 σ |Dσ| dx ≤ C σ p+1 (1 + |Du|2 ) dx k Tn Tn  + C(p + 1) 17

Tn

σ p |Dσ|(1 + |Du|2 ) dx.

Hence 

 σ

(5.5)

p−1

Tn

|Dσ| dx ≤ Ck 2

2 Tn

σ p+1 (1 + |Du|4 ) dx.

Sobolev’s inequality provides the estimate 1  1+θ

 σ

(p+1)(1+θ)

 ≤C

dx

Tn

Tn

|Dσ

p+1 2

|2 + σ p+1 dx 

 ≤ C(p + 1)

2

σ

p−1

Tn

|Dσ| dx + C 2

σ p+1 dx, Tn

where θ > 0 is the same as in the proof of Lemma 3.1. Therefore (5.5) implies 1  1+θ

 σ

(5.6)

(p+1)(1+θ)

 ≤ C(p + 1) k

dx

2 2

Tn

Tn

σ p+1 (1 + |Du|4 ) dx.

Write β := (1 + θ)1/2 > 1 and calculate ¯ 2+ ) ≤ C(1 + σ |Du|4 ≤ C(1 + h2 ) ≤ C(1 + (h − H)

β−1 β

).

Consequently,  σ

(p+1)β

2



1 β2

dx

Tn

 ≤ C(p + 1) k

2 2

σ p+1 (1 + σ

β−1 β

) dx.

Tn

Since  σ

p+1

σ



β−1 β

dx ≤

Tn

we deduce

 σ

1/β  σ

(p+1)β

1− β1

dx

σ dx

Tn

 =

Tn

1/β 2

(p+1)β 2

1/β

 ≤ C(p + 1) k

dx

Tn

σ

(p+1)β

Tn

that is, 1/β

 σ dx

(5.7) Tn

σ Tn

2 2

qβ

1/β

 ≤ Cq k

2β 2β

for each q = (p + 1)β ≥ β. 18

(p+1)β

σ q dx Tn

dx

;

dx

,

3. We next employ Moser’s method of estimating higher and higher Lp norms. For this, set q = qm := β m in (5.7): 1/qm+1

 σ

qm+1

≤C

dx

Tn

1 qm

2β qm

qm k

2β qm

1/qm

 σ

qm

dx

,

Tn

for m = 1, . . . . Iterating, we find σL∞ (Tn ) ≤ C



1 βm



β

2m β m−1



k

2 β m−1

1/β

 β

σ dx Tn

≤C k

3β−1 β−1

,

¯

where we used (3.4) to estimate the next-to-last term. But since σ = ek(h−H) , this implies ¯ = log σ ≤ log C + 3β − 1 log k, k(h − H) β−1 and so

¯ + C log k h≤H k


for another constant C.

Solving the Euler–Lagrange equations. As an application, we discuss finding smooth solutions of the Euler–Lagrange PDE (1.7). Theorem 5.2. For each k = 1, . . . , there exists a unique, smooth solution uk = P · x + v k  of the Euler–Lagrange equation (1.7), where v k is Tn –periodic and Tn v k dx = 0. Proof. 1. Define for each 0 ≤ λ ≤ 1 the Hamiltonian Hλ (p, x) := λH(p, x) + (1 − λ)

|p|2 , 2

for p, x ∈ Rn ; and for each k introduce the PDE (5.8λ )

div(ekHλ (Du,x) Dp Hλ (Du, x)) = 0

 where u = P · x + v, v being Tn –periodic with Tn v dx = 0. Define Λ := {λ | 0 ≤ λ ≤ 1, (5.8λ ) has a smooth solution}. Clearly we have 0 ∈ Λ, corresponding to the solution u = P · x; that is, v ≡ 0. 2. We assert next that Λ is closed. This is a consequence of Theorem 5.1, since the Hamiltonians Hλ each satisfy the basic assumptions. Consequently, we can bound the first derivatives of solutions of (5.8λ ), and elliptic regularity theory implies we can then estimate their derivatives of any order. These bounds are independent of λ ∈ Λ. 19

So any convergent subsequence of elements of Λ contains a further subsequence whose corresponding solutions of the PDE converge uniformly, along with all derivatives. 3. Finally, we claim that Λ is open. To see this, take any λ ∈ Λ and write out the linearization of (5.8λ ) about the smooth solution u = P · x + v: Lw := −(ekHλ (Du,x) (Hλ,pi pj (Du, x) + kHλ,pi (Du, x)Hλ,pj (Du, x))wxj )xi . This is a symmetric, uniformly elliptic operator, whose null space among periodic functions consists of the constants. Using the Implicit Function Theorem, we construct a unique solution for nearby values of λ. 4. Since Λ is nonempty, closed and open, we have Λ = [0, 1] and in particular 1 ∈ Λ. Thus the Euler–Lagrange PDE has a smooth solution u = P · x + v. It is unique since any such solution gives the unique minimizer of Ik [·], subject to the requirement that  v dx = 0.
Tn 6. Appendix B: A quick proof of the min–max formula. The following argument is due to A. Fathi. ¯ ) and let σ be a corresponding Mather Let u be a viscosity solution of H(Du, x) = H(P ˆ = P · x + vˆ, where vˆ any Tn -periodic, C 1 measure, so that div(σDp H(Du, x)) = 0. Take u function. Then convexity implies u − Du) ≤ H(Dˆ u, x). H(Du, x) + Dp H(Du, x) · (Dˆ We integrate with respect to σ and deduce thereby that   ¯ H(Du, x) dσ ≤ H(Dˆ u, x) dσ ≤ maxn H(Dˆ u, x). H(P ) = Tn

Tn

x∈T

¯ ) ≤ inf vˆ∈C 1 maxx∈Tn H(P + Dˆ v , x). Thus H(P On the other hand, if ηδ is a radial convolution kernel, the smoothed function uδ := ηδ ∗ u = P · x + vδ satisfies ¯ ) + O(δ) H(Duδ , x) ≤ H(P ¯ ). v , x) ≤ H(P on Tn . Therefore inf vˆ∈C 1 maxx∈Tn H(P + Dˆ

20




References G. Aronsson, Minimization problems for the functional sup F (x, f (x), f
(x)), Ark. Mat. 6 (1965), 33–53. [A-2] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561. [B] E. N. Barron, Viscosity solutions and analysis in L∞ , in Nonlinear Analysis, Differential Equations and Control (1999), Dordrecht, 1–60. [B-J-W] E. N. Barron, R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of L∞ functionals, Arch. Ration. Mech. Analysis 157 (2001), 255–283. [C-I-P] G. Contreras, R. Iturriaga and G. P. Paternain, Lagrangian graphs, minimizing measures and Mane’s critical values, Geom. Funct. Analysis 8 (1998), 788–809. [D-E] D. M. Duc and J. Eells, Regularity of exponentially harmonic functions, Internat. J. Math 2 (1991), 395–408. [E-G] L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis 157 (2001), 1–33. [F1] A. Fathi, Th´ eor` eme KAM faible et th´ eorie de Mather sur les syst` emes lagrangiens, C. R. Acad. Sci. Paris Sr. I Math. 324 (1997), 1043–1046. [F2] A. Fathi, Solutions KAM faibles conjugu´ ees et barri` eres de Peierls, C. R. Acad. Sci. Paris Sr. I Math. 325 (1997), 649–652. [F3] A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version, Lecture notes (2001). [L] G. Lieberman, On the regularity of the minimizer of a functional with exponential growth, Comm. Math. Univ. Carol. 33 (1992), 45–49. [L-P-V] P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton–Jacobi equations, unpublished, circa 1988. [Mt1] J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989), 375–394. [Mt2] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Zeitschrift 207 (1991), 169–207. [M-F] J. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math 1589 (S. Graffi, ed.), Sringer, 1994. [N] H. Naito, On local H¨ older continuity for a minimizer of the exponential energy functional, Nagoya Math. J. 129 (1993), 97–113. [P] G. P. Paternain, Schr¨ odinger operators with magnetic fields and minimal action functionals, to appear in Israel Journal of Math.

[A-1]

Department of Mathematics, University of California, Berkeley, CA 94720

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