ON BOUNDARY BLOW-UP PROBLEMS FOR THE COMPLEX ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 12, December 2008, Pages 4355–4364 S 0002-9939(08)09513-0 Article electronically published on July 8, 2008

ON BOUNDARY BLOW-UP PROBLEMS ` FOR THE COMPLEX MONGE-AMPERE EQUATION ´ SZYMON PLIS (Communicated by Mei-Chi Shaw) Abstract. We prove the C ∞ regularity for some complex Monge-Amp`ere equations with boundary data equal to +∞.

1. Introduction Cheng and Yau in [C-Y] considered the following problem:
det(up¯q ) = geKu in Ω, (1.1) limz→z0 u(z) = +∞ for every z0 ∈ ∂Ω, where Ω is a strictly pseudoconvex domain with C ∞ boundary (or some more general ¯ is a positive function. They non-compact K¨ahler manifold), K > 0 and g ∈ C ∞ (Ω) proved that problem (1.1) has a unique plurisubharmonic solution u of class C ∞ . This immediately gives a complete K¨ahler-Einstein metric on Ω. We will consider a more general problem than (1.1), i.e.
det(up¯q ) = gf (u) in Ω, (1.2) limz→z0 u(z) = +∞ for every z0 ∈ ∂Ω, where Ω and g are as above, and f ∈ C(R) ∩ C ∞ ((τ, +∞)) for some τ ∈ R ∪ {−∞} such that f = 0 on (−∞, τ ) and f  > 0 on (τ, +∞). We will show (see Theorem 2.1) the existence of a smooth solution for functions f satisfying some technical et ··

e·

with τ = −∞ and if conditions, which are fulfilled for example if f (t) = e f (t) = tp for p > n, f (t) = tn + tn+1 , and f (t) = tn (log(t + 2))2n with τ = 0. This article is organized as follows. In section 2 we formulate the main theorem, and we give the proof of it in sections 3-5. In sections 6 and 7 we obtain something more about the case f (t) = tp . 2. Main theorem Let us first consider the following condition: W1: the function  ∞ dt Ψ(x) = , 1/(n+1) F (t) x Received by the editors November 6, 2007. 2000 Mathematics Subject Classification. Primary 32W20, 35B65. Key words and phrases. Complex Monge-Amp` ere equation, blow up problem. This research was partially supported by Polish grant MNiSW 3342/H03/2006/31. c 2008 American Mathematical Society Reverts to public domain 28 years from publication

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where F is such that F  = f , is well defined for x large enough. It is easy to check (see [G-P]) that condition W1 implies F (x)n/(n+1) = 0. x→+∞ f (x)

(2.1)

lim

Ivarsson and Matero proved in [I-M] that if Ω = B = {z : |z| < 1}, g depends only on |z|, τ = −∞ and f satisfies condition W1, then problem (1.2) has a plurisubharmonic solution u ∈ C ∞ (B). For Ω strictly pseudoconvex with the boundary of class C ∞ Ivarsson proved in [I] that if τ = −∞, g ≡ 1, f fulfills W1
(x) ≥ n−1 and the condition F (x)f f 2 (x) n+1 for x large enough, then problem (1.2) has a plurisubharmonic and locally Lipschitz solution u. By [C-K-N-S] there exists a unique solution w of ⎧ ¯ ⎨ w ∈ PSH(Ω) ∩ C ∞ (Ω), det(wp¯q ) = g in Ω, ⎩ w = 0 on ∂Ω. Let us put



∞

Φ(x) = x

dt for x > τ ; f (t)1/n

then by the condition W1, it is well defined. Let T = (Φ−1 (supΩ (−w)), +∞). To formulate the main theorem we will also consider the following two conditions: W2: Φ(τ ) = lim+ Φ(x) > sup(−w). x→τ

Ω

W3: There exist α, β, γ ≥ 1 such that on T we have α−1 f −β ≤ f  ≤ αf β , (log f ) ≤ γ(log f )2 . Now we can formulate the main theorem: Theorem 2.1. Let Ω, f and g be as above. If W1-W3 are satisfied, then there exists a plurisubharmonic solution u ∈ C ∞ (Ω) of the problem (1.2). Note that if τ = −∞ or f (τ +x) ≤ Cxn for x in some interval [0, ε], then condition W2 is satisfied. Otherwise, for ε small enough it is satisfied in εΩ = {εz : z ∈ Ω} if we also change g to g ◦ S, where S(z) = zε . Theorem 3.1 in [M] claims the existence of a smooth solution to a similar problem for the real Monge-Amp`ere equation without any condition on f near τ . We believe however that the proof only works if f is replaced by λf , where λ is a constant such that the condition analogous to W2 holds. So it seems to be natural (in the complex and real cases) to assume something like W2. We will see that it enables us to construct a solution u > τ . We believe that assumption W3 in Theorem 2.1 is in fact superfluous. However the author was unable to prove a priori estimates without this condition. To see that many functions satisfy this condition, note that for positive and increasing f and x > 1 we have lim inf t→+∞

f  (t) (log f (t)) = 0 ≥ lim inf . t→+∞ (log f (t))x f (t)x

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` BLOW-UP PROBLEMS FOR COMPLEX MONGE-AMPERE EQUATION

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Also, if f satisfy W 1, then lim sup t→+∞

f  (t) = +∞. f (t)n−1/n

3. Approximating sequence In the same way as in [M] (in the real case) we will define an increasing sequence of plurisubharmonic functions approximating the solution of problem (1.2). Let ϕ = (−Φ)−1 ◦ w, Ωk = {z ∈ Cn : ϕ(z) < k} (where k is such that the set Ωk is a strictly pseudoconvex domain of class C ∞ ) and let uk be the solution of the following Dirichlet problem: ⎧ ¯ k ), ⎨ u ∈ PSH ∩ C ∞ (Ω det(up¯q ) = gf (u) in Ωk , ⎩ u = k(= ϕ) in ∂Ωk . The following lemma is an easy consequence of the comparison principle. Lemma 3.1. We have uk+1 ≥ uk ≥ ϕ. Proof. Let V = (−Ψ)−1 . Then V > 0 is smooth, V  = f 1/n ◦ V > 0 and V  = f
◦V > 0. So V is a smooth plurisubharmonic function, and we can compute nF (n−2)/n det(ϕp¯q ) = det(V  ◦ wwp¯q + V  ◦ wwp wq¯) 1 = gf ◦ V ◦ w + gf 1/n f  ◦ V ◦ wwp¯q wp wq¯ > gf ◦ ϕ. n By the comparison principle uk ≥ ϕ, and from this (again using the comparison
principle) we obtain uk+1 ≥ uk . 4. A priori estimates In this section we shall prove that uk and uk are locally bounded, and our estimate will be independent of k. Differentiating (the logarithm of) the equation det(up¯q ) = gf (u) twice we get the following formulas, which are very useful in the theory of regularity of the Monge-Amp`ere equation: (4.1)

(log gf (u))k = up¯q ukp¯q ,

(4.2)

pj i¯ q (log gf (u))kk¯ = up¯q ukkp¯ ¯ q − u u uki¯ ¯ q, j ukp¯

¯

where k = 1, . . . , n and (up¯q ) is the inverse transposed matrix of (up¯q ). We will q ∂2 also use the differential operator L = Lk = up¯ k ∂zp ∂ z¯q . Functions uk are plurisubharmonic; hence L is elliptic and we can apply the maximum principle. In a similar way as in [I] (or in the real case in [M]) we will prove the following lemma: Lemma 4.1. There is a function H ∈ PSH ∩ C(Ω), such that for every u ∈ ¯ if det(up¯q ) ≥ gf (u), then u ≤ H. PSH(Ω) ∩ C(Ω),

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Proof. Let W = (−Ψ)−1 and H = W ◦(Kw), where K > 0 (small enough so that H is well defined) will be specified later. Then W > 0 is smooth, W  = F 1/(n+1) ◦W > f ◦W > 0. So H is a smooth plurisubharmonic function, 0 and W  = (n+1)F (n−1)/(n+1) and we can compute det(Hp¯q ) = det(W  ◦ (Kw)Kwp¯q + W  ◦ (Kw)K 2 wp wq¯) 1 gK n+1 f ◦ W ◦ (Kw)wp¯q wp wq¯ = gK n F n/(n+1) ◦ W ◦ (Kw) + n+1  n n/(n+1)  K F wp¯q wp wq¯ ◦ H + K n+1 = gf (H) . f n+1 By (2.1), for K small enough we have det(Hp¯q ) ≤ gf (H). So by the comparison principle we obtain u ≤ H.
Lemma 4.2. There exists a constant C independent of k (it may depend on Ω, f and g) such that (4.3)

|∇uk | ≤ Cf 1/n (k).

Proof. Since uk ≥ ϕ, on ∂Ωk we have (as in [I]) f −1/n (uk )|∇uk | = |∇(Φ ◦ uk )| ≤ |∇(Φ ◦ ϕ)| = |∇w|. Let r ∈ {1, . . . , 2n}. To estimate (uk )xr let us consider the function η = (uk )xr + Kf 1/n (k)|z|2 . If η attains a maximum on the boundary, then the proof is complete, so we may assume that η attains maximum at z0 ∈ Ωk . We can also assume that (uk )xr (z0 ) ≥ 0. By (4.1) and because f is increasing, we have at z0 L((uk )xr + Kf 1/n (k)|z|2 ) = (log(gf (uk )))xr + Kf 1/n (k)

n 

up¯p

p=1

≥ (log g)xr + nK > 0 for K large enough. By the maximum principle we obtain (4.3).




Lemma 4.3. If conditions W1–W3 are fulfilled, then there are constants N and C independent of k such that uk ≤ C on Ωk . f (uk )N Proof. Let c0 , c1 , c2 , . . . be constants depending only on Ω, wC 2,1 , f and g. For simplicity in what follows we will write u instead of uk , Ω instead of Ωk and η instead of η ◦ u (or η(u)). We will show that for N big enough the function Λ=

max i∈{1,...,n}

λi f −N e|z| , 2

where λi are eigenvalues of the matrix (up¯q ), is bounded . The proof will be divided into two parts: first we will estimate Λ on the boundary and then inside.

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Part I: Function Λ is bounded on boundary of Ω. Let z0 ∈ ∂Ω. After a holomorphic change of variables (see for example the proof of Lemma 1.3 in [C-K-N-S]) we can assume that z0 = 0, w = −x2n−1 + ap¯q zp z¯q + o(|z|2 ) for some matrix (ap¯q )np,q=1 > 0, wp¯q = ap¯q = 0 for p, q < n such that p = q. Then (as in [C-K-N-S]) (4.4)

ux2n−1 wp¯q for p, q < n. wx2n−1

up¯q =

For l = 1, . . . , 2n − 2 we consider wl = ±Tl u + (uxl )2 + (ux2n )2 + A|z|2 − Bx2n−1 , where Tl = a∂xl − b∂x2n−1 , a = ρx2n−1 and b = ρxl . We have up¯q ux2n−1 q¯ = 2up¯q un¯q + iup¯q ux2n q¯ = 2δpn + iup¯q ux2n q¯ , up¯q ux2n−1 p = 2up¯q up¯n − iup¯q ux2n p = 2δqn − iup¯q ux2n p . Thus using (4.1) and the Schwarz inequality we can calculate Lwl = ±up¯q (ap¯q uxl − bp¯q ux2n−1 + ap uxl q¯ − bp ux2n−1 q¯ + aq¯uxn p − bq¯ux2n−1 p ) ± Tl log(gf ) + 2uxl (log(gf ))xl + 2up¯q upxl uq¯xl  + 2ux2n (log(gf ))x2n + 2up¯q upx2n uq¯x2n + A up¯p ≥ −c1 |∇u|



u

p¯ p

p

− c2



p

u

p¯ p

− 2 up¯q ap aq¯ up¯q uxl p uxl q¯

p

f  |∇u| f  |∇u|2 − c5 |∇u| − 4 − 2 up¯q bp bq¯ up¯q ux2n p ux2n q¯ − c3 − c4 f f  p¯ q p¯ q p¯ p + 2u upxl uq¯xl + 2u upx2n uq¯x2n + A u . p

We have



up¯p ≥

1 ; f 1/n (k)

hence (also using the inequality between arithmetic and

geometric means and Lemma 4.2) for some A < c6 (f 2/n (k) + supΩ f  ) we obtain Lwl > 0. Let Sε be the connected component of the set {z ∈ Ω : x2n−1 < ε} such that 0 ∈ S¯ε . For well chosen (small enough and independent of k and z0 ) ε > 0 and for every z ∈ ∂Sε we have c7 √ √ |z| ≤ x2n−1 = c8 x2n−1 . ε Then, since Tl u = 0 on ∂Ω ∩ ∂Sε , we have ±Tl u ≤ c9 x2n−1 on ∂Sε , and by Lemma 4.2 (uxr )2 ≤ c10 |∇u|2 |z|2 ≤ c11 f 2/n (k)x2n−1 for r = 2n − 1.

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Hence choosing suitable B < c12 (f 2/n (k) + supΩ f  ) we obtain wl ≤ 0 on ∂Sε . From the maximum principle wl ≤ 0 holds on the whole of Sε . This implies that wlx2n−1 < c13 (f 2/n (k) + supΩ f  ). By Lemma 4.2 and condition W3 we obtain (4.5)

|up¯n | < c14 f N for some N > 0.

In our coordinates at 0 we can write the Monge-Amp`ere equation in the form    gf = ukk¯ − up¯n unp¯ ukk¯ . k

p=n

p=k=n

Solving this and using (4.4) we can calculate (4.6)

un¯n =

 up¯n unp¯wx2n−1  f gux2n−1 − . wp¯p ux2n−1 wkk¯ ux2n−1

p=n

k=n

By Lemma 3.1 there exists a constant ν > 0 such that f g > ν on Ω. So using the inequality between arithmetic and geometric means, we can estimate u ≥ nν 1/n , and then from the Hopf Lemma there exists a constant µ > 0 such that ux2n−1 > µ. Thus, using (4.4), (4.5) and (4.6) we conclude that u < c15 f N . ¯ Assume that a maximum of the function Part II: Function Λ is bounded in Ω. Λ is attained at z0 ∈ Ω (otherwise we are done). After a linear change of variables we can assume that at z0 the matrix (up¯q ) is diagonal and u1¯1 = maxk∈{1,...,n} λk . 2 Let h = u1¯1 f −N e|z| . Then the function h also attains a maximum at z0 and h(z0 ) = Λ(z0 ). From now on all formulas are assumed to hold at z0 . We may assume that h ≥ Ce|z0 |

2

(4.7)

for some (big enough) C > 0, which will be specified later, and that |z0 |
1. The Convergence Theorem for the complex Monge-Amp`ere equation gives us the fact that u is a solution of problem (1.2), so using Theorem 2.5 from [B1] we get u ∈ C 2 (Ω). Now Theorem 2.1 follows from Theorem 2.2 (again) from [B1].
Note that in Theorem 2.1 we can (without changing the proof) assume that ¯ ∩ C ∞ (Ω) instead of smoothness of ∂Ω. w ∈ C 2,1 (Ω) 6. A necessary condition Similar to the real case (see [L-M]) for the function f (t) = tp , problem (1.2) has a solution if and only if p > n. It follows from the following theorem: Theorem 6.1. If a solution of the problem ⎧ ⎨ u ∈ PSH ∩ C(Ω), det(up¯q ) ≤ gf (u) in Ω, (6.1) ⎩ limz→z0 u(z) = +∞ for z0 ∈ ∂Ω  1 exists, then the integral f 1/n converges at +∞. Proof (similar to [M]). We can assume that τ < 0 and that the function  x dt µ(x) = 1/n (t) f 0 is not bounded. Let u be a solution of (6.1). Then for A big enough the function φA = µ−1 ◦ (w + A) is well defined. In the same way as in Lemma 3.1, for the

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function ϕ we obtain det(φAp¯q ) ≥ gf (φA ). From the comparison principle u ≥ φA . It is a contradiction with a free choice of large A.
7. Uniqueness Ivarsson in [I] proved that if lim supt→+∞ −Ψ(t) Ψ
(t) < +∞, then the solution of problem (1.2) is unique. Using his method, we will also show this for f (t) = tp where p > n (in this case limt→+∞ −Ψ(t) Ψ
(t) = +∞). Theorem 7.1. Let u, v be plurisubharmonic solutions of problem (1.2). Assume τ = −∞, W1, (7.1)

lim sup t→+∞

−Ψ(t) < +∞ tΨ (t)

and also n t+C for some constant C > 0 and t ≥ inf Ω min{u, v}. Then u ≡ v. (log f (t)) ≥

(7.2)

Proof. First, we will prove that (7.3)

lim

z→∂Ω

u(z) = 1. v(z)

Indeed, using (7.1), we obtain that there exist D > 0 such that for t big enough and x ≥ 1 we have ∂ (Ψ(xt)) = xΨ (xt) ≤ −DΨ(xt), ∂x so (7.4)

Ψ(xt) ≤ Ψ(t)(eD )1−x .

Theorem 3.8 from [I-M] gives us Ψ(u(z)) = 1. z→∂Ω Ψ(v(z)) lim

Thus, using (7.4) we can conclude that (7.3) follows. Let x ≥ 1, ux = x(u + C), vx = x(v + C) and fx (t) = xn f ( xt − C) for t ∈ R. Note that ux , vx > 0 and det(ux p¯q ) = det(vx p¯q ) = gfx (u) on Ω. Inequality (7.2) x (t) means that ∂f∂x ≥ 0 for t ∈ R, so fx ≥ f1 . Using (7.3) we obtain that the set S = {ux < v1 } is relatively compact in Ω for x > 1. Then by the comparison principle S is empty. Hence, because limx→1+ ux = u1 , we have u1 ≥ v1 , and in
the same way we can obtain that v1 ≥ u1 . Thus the theorem follows. Corollary 7.2. Let p > n and f (t) = tp . Then there is exactly one plurisubharmonic solution of problem (1.2). Proof. By Theorem 2.1 we know that there is a smooth solution of problem (1.2). The definition of ϕ and Lemma 3.1 imply that our solution is positive. If u is a positive solution we may suitably modify the function f (t) = tp for t < inf Ω u and get a positive, increasing function on R. Hence from Theorem 7.1 we have only one positive solution. Now it is enough to prove that every solution is positive. To ˜ such that Ω is relatively do this let us consider a strictly pseudoconvex domain Ω,

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´ SZYMON PLIS

˜ Then on Ω ˜ there exists a positive plurisubharmonic solution u of compact in Ω. the equation det(up¯q ) = (sup g)up , Ω

and from the comparison principle u is dominated from above by any solution of problem (1.2) which gives the statement.
Acknowledgment The author would like to express his gratitude to Z. Blocki for helpful discussions and advice during the work on this paper. References Z. B
locki, On the regularity of the complex Monge-Amp` ere operator, Complex geometric analysis in Pohang (1997), 181-189, Contemp. Math., 222, Amer. Math. Soc., Providence, RI, 1999. MR1653050 (99m:32018) [B2] Z. B
locki, Regularity of the degenerate Monge-Amp` ere equation on compact K¨ ahler manifolds, Math. Z. 244 (2003), no. 1, 153-161. MR1981880 (2004b:32065) [C-K-N-S] L. Caffarelli, J. J. Kohn, L. Nirenberg, J. Spruck, The Dirichlet problem for non-linear second order elliptic equations, II: Complex Monge-Amp` ere, and uniformly elliptic equations, Comm. Pure Appl. Math. 38 (1985), 209-252. MR780073 (87f:35097) [C-Y] S.-Y. Cheng, S.-Y. Yau, On the existence of a complete K¨ ahler metric on non-compact complex manifolds and regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980), 507-544. MR575736 (82f:53074) [G-P] F. Gladiali, G. Porru, Estimates for explosive solutions to p-Laplace equations, Progress in Partial Differential Equations (Pont-` a-Mousson), Vol. 1, Pitman Res. Notes Math. Series, 383, Longman, Harlow (1998), 117-127. MR1628068 (2000h:35047) [I] B. Ivarsson, Regularity and uniqueness of solutions to boundary blow-up problems for the complex Monge-Amp` ere operator, Bull. Polish Acad. Sci. Math. 54 (2006), 13-25. MR2270791 (2007g:32028) [I-M] B. Ivarsson, J. Matero, The blow-up rate of solutions to boundary blow-up problems for the complex Monge-Amp` ere operator, Manuscripta Math. 120 (2006), no. 3, 325-345. MR2243567 (2007h:32058) [L-M] A. C. Lazer, P. J. McKenna, On singular boundary value problems for the MongeAmp` ere operator, J. Math. Anal. Appl. 197 (1996), 341-362. MR1372183 (97c:35064) [M] A. Mohammed, On the existence of solutions to the Monge-Amp` ere equation with infinite boundary values, Proc. Amer. Math. Soc. 138 no. 1 (2007), 141-149. MR2280183 [B1]

Institute of Mathematics, Cracow University of Technology, Warszawska 24, 31-155 ´ w, Poland Krako E-mail address: [email protected]

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