BOUNDARY REGULARITY FOR SOLUTIONS TO THE LINEARIZED MONGE-AMPÈRE EQUATIONS

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1 BOUNDARY REGULARITY FOR SOLUTIONS TO THE LINEARIZED MONGE-AMPÈRE EQUATIONS N. Q. LE AND O. SAVIN Abstract. We obtain boundary Hölder gradient estimates and regularity for solutions to te linearized Monge-Ampère equations under natural assumptions on te domain, Monge-Ampère measures and boundary data. Our results are affine invariant analogues of te boundary Hölder gradient estimates of Krylov. 1. Introduction Tis paper is concerned wit boundary regularity for solutions to te linearized Monge- Ampère equations. Te equations we are interested in are of te form wit L u v := L u v = g, n U ij v ij, i,j=1 were u is a locally uniformly convex function and U ij is te cofactor of te Hessian D 2 u. Te operator L u appears in several contexts including affine differential geometry [TW, TW1, TW2, TW3], complex geometry [D2], and fluid mecanics [B, CNP, Loe]. As U = (U ij ) is divergence-free, we can write n n L u v = i (U ij D j v) = i j (U ij v). i,j Because te matrix of cofactors U is positive semi-definite, L u is a linear elliptic partial differential operator, possibly degenerate. In [CG], Caffarelli and Gutiérrez developed a Harnack inequality teory for solutions of te omogeneous equations L u v = 0 in terms of te pincing of te Hessian determinant i,j=1 λ det D 2 u Λ. Tis teory is an affine invariant version of te classical Harnack inequality for uniformly elliptic equations wit measurable coefficients. In tis paper, we establis boundary Hölder gradient estimates and regularity for solutions to te linearized Monge-Ampère equations L u v = g under natural assumptions on te domain, Monge-Ampère measures and boundary data; see Teorems 2.1, 2.4 and 2.5. Tese teorems are affine invariant analogues of te boundary Hölder gradient estimates of Krylov [K]. 1

2 2 N. Q. LE AND O. SAVIN Te motivation for our estimates comes from te study of convex minimizers u for convex energies E of te type E(u) = F (det D 2 u) dx + udσ uda, Ω wic we considered in [LS2]. Suc energies appear in te work of Donaldson [D1]-[D4] in te context of existence of Käler metrics of constant scalar curvature for toric varieties. Minimizers of E satisfy a system of te form F (det D 2 u) = v in Ω, U ij v ij = da in Ω, (1.1) v = 0 on Ω, U νν v ν = σ on Ω, were U νν = det D 2 x u wit x ν denoting te tangential directions along Ω. Te minimizer u solves a fourt order elliptic equation wit two nonstandard boundary conditions involving te second and tird order derivatives of u. In [LS2] we apply te boundary Hölder gradient estimates establised in tis paper and sow tat u C 2,α (Ω) in dimensions n = 2 under suitable conditions on te function F and te measures da and dσ. Our boundary Hölder gradient estimates depend only on te bounds on te Hessian determinant det D 2 u, te quadratic separations of u from its tangent planes on te boundary Ω and te geometry of Ω. Under tese assumptions, te linearized Monge-Ampère operator L u is in general not uniformly elliptic, i.e., te eigenvalues of U = (U ij ) are not necessarily bounded away from 0 and. Moreover, L u can be possibly singular near te boundary; even if det D 2 u is constant in Ω, U can blow up logaritmically at te boundary, see Proposition 2.6. Te degeneracy and singularity of L u are te main difficulties in establising our boundary regularity results. We andle te degeneracy of L u by working as in [CG] wit sections of solutions to te Monge-Ampère equations. Tese sections ave te same role as euclidean balls ave in te classical teory. To overcome te singularity of L u near te boundary, we use a Localization Teorem at te boundary for solutions to te Monge-Ampère equations wic was obtained in [S, S2]. Te rest of te paper is organized as follows. We state our main results in Section 2. In Section 3, we discuss te Localization Teorem and weak Harnack inequality, wic are te main tools used in te proof of our local boundary regularity result, Teorem 2.1. In Sections 4 and 5, we study boundary beavior and te main properties of te rescaled functions u obtained from te Localization Teorem. Te proofs of Teorems 2.1 and 2.5 will be given in Section 6 and Section 7. Let Ω R n be a bounded convex set wit Ω 2. Statement of te main results (2.1) B ρ (ρe n ) Ω {x n 0} B 1, ρ Ω

3 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 3 for some small ρ > 0. Assume tat (2.2) Ω contains an interior ball of radius ρ tangent to Ω at eac point on Ω B ρ. Let u : Ω R, u C 0,1 (Ω) C 2 (Ω) be a convex function satisfying (2.3) det D 2 u = f, 0 < λ f Λ in Ω. Trougout, we denote by U = (U ij ) te matrix of cofactors of te Hessian matrix D 2 u, i.e., U = (det D 2 u)(d 2 u) 1. We assume tat on Ω B ρ, u separates quadratically from its tangent planes on Ω. Precisely we assume tat if x 0 Ω B ρ ten (2.4) ρ x x 0 2 u(x) u(x 0 ) u(x 0 )(x x 0 ) ρ 1 x x 0 2, for all x Ω. Wen x 0 Ω, te term u(x 0 ) is understood in te sense tat x n+1 = u(x 0 ) + u(x 0 ) (x x 0 ) is a supporting yperplane for te grap of u but for any ε > 0, x n+1 = u(x 0 ) + ( u(x 0 ) εν x0 ) (x x 0 ) is not a supporting yperplane, were ν x0 denotes te exterior unit normal to Ω at x 0. In fact we will sow in Proposition 4.1 tat our ypoteses imply tat u is always differentiable at x 0 and ten u(x 0 ) is defined also in te classical sense. We are ready to state our main teorem. Teorem 2.1. Assume u and Ω satisfy te assumptions (2.1)-(2.4) above. Let v : B ρ Ω R be a continuous solution to { U ij v ij = g in B ρ Ω, v = 0 on Ω B ρ, Ten v ν C 0,α ( Ω B ρ/2 ) C ( v L (Ω B ρ) + g/ tr U L (Ω B ρ) ), and, for r ρ/2, we ave te estimate ) max v + v ν (0)x n Cr ( v 1+α L (Ω B + g/ tr U ρ) L (Ω B ρ), B r Ω were α (0, 1) and C are constants depending only on n, ρ, λ, Λ. We remark tat our estimates do not depend on te C 0,1 (Ω) norm of u or te smootness of u. Remark 2.2. Te teorem is still valid if we consider te equation tr (AD 2 v) = g, wit 0 < λu A ΛU and ten te constants α, C depend also on λ, Λ.

4 4 N. Q. LE AND O. SAVIN Teorem 2.1 is concerned wit boundary regularity in te case wen te potential u is nondegenerate along Ω. It is an affine invariant analogue of te boundary Hölder gradient estimate of Krylov [K]. Teorem 2.3 (Krylov). Let w C(B + 1 ) C 2 (B + 1 ) satisfy Lw = f in B + 1, w = 0 on {x n = 0}, were L = a ij ij is a uniformly elliptic operator wit bounded measurable coefficients wit ellipticity constants λ, Λ. Ten tere are constants 0 < α < 1 and C > 0 depending on λ, Λ, n suc tat w n C α (B 1/2 {x n=0}) C( w L (B + 1 ) + f L (B + 1 )). We also obtain global boundary regularity estimates under global conditions on te domain Ω and te potential function u. Teorem 2.4. Assume tat Ω B 1/ρ contains an interior ball of radius ρ tangent to Ω at eac point on Ω. Assume furter tat det D 2 u = f wit λ f Λ, and on Ω, u separates quadratically from its tangent planes, namely ρ x x 0 2 u(x) u(x 0 ) u(x 0 )(x x 0 ) ρ 1 x x 0 2, x, x 0 Ω. Let v : Ω R be a continuous function tat solves { U ij v ij = g in Ω, v = ϕ on Ω, were ϕ is a C 1,1 function defined on Ω. Ten and for all x 0 Ω v ν C 0,α ( Ω) C ( ϕ C 1,1 ( Ω) + g/ tr U L (Ω)), max v v(x 0 ) v(x 0 )(x x 0 ) C ( ϕ C 1,1 ( Ω) + g/ tr U L (Ω)) r 1+α, B r(x 0 ) Ω were α (0, 1) and C are constants depending on n, ρ, λ, Λ. Teorem 2.4 follows easily from Teorem 2.1. Indeed, first we notice tat v is bounded by te use of barriers ±C( x 2 2/ρ 2 ), for appropriate C, and ten we apply Teorem 2.1 on Ω for ṽ := v ϕ, were ϕ is a C 1,1 extension of ϕ to Ω. If, in addition, we assume tat det D 2 u is globally Hölder continuous, ten te solutions to te linearized Monge-Ampère equations ave global C 1,α estimates as stated in te next teorem.

5 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 5 Teorem 2.5. Assume te ypoteses of Teorem 2.4 old and f C β (Ω) for some β > 0. Ten v C 1,α (Ω) K( ϕ C 1,1 ( Ω) + g/ tr U L (Ω)), wit K a constants depending on n, β, ρ, λ, Λ and f C β (Ω). Finally we mention also te regularity properties of te potentials u tat satisfy our ypoteses. Proposition 2.6. If u satisfies te ypoteses of Teorem 2.4 ten If in addition f C β (Ω) ten [ u] C α (Ω) C. D 2 u K log ε 2 on Ω ε = {x Ω, dist(x, Ω) > ε}, were K is a constant depending on n, β, ρ, λ, Λ and f C β (Ω). Te proof of Teorem 2.1 follows te same lines as te proof of te standard boundary estimate of Krylov. Our main tools are a localization teorem at te boundary for solutions to te Monge-Ampère equation wic was obtained in [S], and te interior Harnack estimates for solutions to te linearized Monge-Ampère equations wic were establised in [CG] (see Section 3). 3. Te Localization Teorem and Weak Harnack Inequality In tis section, we state te main tools used in te proof of Teorem 2.1, te localization teorem and te weak Harnack inequality. We start wit te localization teorem. Let u : Ω R be a continuous convex function and assume tat (3.1) u(0) = 0, u(0) = 0. Let S (u) be te section of u at 0 wit level : S := {x Ω : u(x) < }. If te boundary data as quadratic growt near {x n = 0} ten, as 0, S is equivalent to a alf-ellipsoid centered at 0. Tis is te content of te Localization Teorem proved in [S, S2]. Precisely, tis teorem reads as follows. Teorem 3.1 (Localization Teorem [S, S2]). Assume tat Ω satisfies (2.1) and u satisfies (2.3), (3.1) above and, (3.2) ρ x 2 u(x) ρ 1 x 2 on Ω {x n ρ}. Ten, for eac < k tere exists an ellipsoid E of volume ω n n/2 suc tat ke Ω S k 1 E Ω.

6 6 N. Q. LE AND O. SAVIN Moreover, te ellipsoid E is obtained from te ball of radius 1/2 by a linear transformation A 1 (sliding along te x n = 0 plane) A E = 1/2 B 1 A (x) = x τ x n, τ = (τ 1, τ 2,..., τ n 1, 0), wit τ k 1 log. Te constant k above depends only on ρ, λ, Λ, n. Te ellipsoid E, or equivalently te linear map A, provides useful information about te beavior of u near te origin. From Teorem 3.1 we also control te sape of sections tat are tangent to Ω at te origin. Before we state tis result we introduce te notation for te section of u centered at x Ω at eigt : S x, (u) := {y Ω : u(y) < u(x) + u(x)(y x) + }. Proposition 3.2. Let u and Ω satisfy te ypoteses of te Localization Teorem 3.1 at te origin. Assume tat for some y Ω te section S y, Ω is tangent to Ω at 0 for some c. Ten tere exists a small constant k 0 > 0 depending on λ, Λ, ρ and n suc tat u(y) = ae n for some a [k 0 1/2, k0 1 1/2 ], k 0 E S y, y k0 1 E, k 0 1/2 dist(y, Ω) k0 1 1/2, wit E te ellipsoid defined in te Localization Teorem 3.1. Proposition 3.2 is a consequence of Teorem 3.1 and was proved [S3]. For completeness we sketc its proof at te end of te paper. Next, we state te weak Harnack inequality. Caffarelli and Gutiérrez [CG] proved Hölder estimates and Harnack inequalities for solutions of te omogeneous equation L u v = 0. Teir approac is based on te Krylov and Safonov s Hölder estimates for linear elliptic equations in general form, wit te sections of u aving te same role as euclidean balls ave in te classical teory. We state te weak Harnack inequality in tis setting (see also [TW3]). Teorem 3.3. (Teorem 4 [CG]) Let u C 2 (Ω) be a locally strictly convex function satisfying 0 < λ det D 2 u Λ, and let v 0 be a nonnegative supersolution defined in a section S x, (u) Ω, L u v := U ij v ij 0. If {v 1} S x, (u) µ S x, (u) ten inf v c, S x,/2 (u) wit c > 0 a constant depending only on n, λ, Λ and µ.

7 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 7 4. Boundary beavior of te rescaled functions We denote by c, C positive constants depending on ρ, λ, Λ, n, and teir values may cange from line to line wenever tere is no possibility of confusion. We refer to suc constants as universal constants. Sometimes, for simplicity of notation, we write S x, instead of S x, (u) and we drop te x subindex wenever x = 0, i.e., S = S 0, (u). We denote te distance from a point x to a closed set Γ as (4.1) d Γ (x) = dist(x, Γ). First we obtain pointwise C 1,α estimates on te boundary in te setting of te Localization Teorem 3.1. We know tat for all k, S satisfies wit A being a linear transformation and ke Ω S k 1 E, det A = 1, E = A 1 B 1/2, A x = x τ x n Tis gives τ e n = 0, A 1, A k 1 log. (4.2) Ω B + c 1/2 / log S B + C 1/2 log, or u in Ω B + c 1/2 / log. Ten for all x close to te origin u(x) C x 2 log x 2, wic sows tat u is differentiable at 0. We remark tat te oter inclusion of (4.2) gives a lower bound for u near te origin (4.3) u(x) c x 2 log x 2 x 3. We summarize te differentiability of u in te next lemma. Lemma 4.1. Assume u and Ω satisfy te ypoteses of te Localization Teorem 3.1 at a point x 0 Ω. If x Ω B r (x 0 ), r 1/2, ten (4.4) u(x) u(x 0 ) u(x 0 )(x x 0 ) Cr 2 log r 2. Moreover, if u, Ω satisfy te ypoteses of te Localization Teorem 3.1 also at a point x 1 Ω B r (x 0 ) ten u(x 1 ) u(x 0 ) Cr log r 2.

8 8 N. Q. LE AND O. SAVIN Clearly, te second statement follows from writing (4.4) for x 0 and x 1 at all points x in a ball B cr (y) Ω. Next we discuss te scaling for our linearized Monge-Ampère equation. Under te linear transformations we find tat ũ(x) = 1 u(t x), a ṽ(x) = 1 v(t x), b g(x) = 1 a n 1 b (det T )2 g(t x), (4.5) Ũ ij ṽ ij = g. Indeed, we note tat D 2 ũ = 1 a T t D 2 ut, D 2 ṽ = 1 b T t D 2 vt, and Ũ = (det D 2 ũ)(d 2 ũ) 1 = 1 a (det T n 1 )2 (det D 2 u) T 1 (D 2 u) 1 (T 1 ) t = 1 a (det T n 1 )2 T 1 U(T 1 ) t and (4.5) easily follows. We use te rescaling above wit a =, b = 1/2, T = 1/2 A 1 were A is te matrix in te Localization teorem. We denote te rescaled functions by and tey satisfy u (x) := u(1/2 A 1 x), v (x) := v(1/2 A 1 x). 1/2 g (x) := 1/2 g( 1/2 A 1 x), (4.6) U ij D ijv = g. Te function u is continuous and is defined in Ω wit and solves te Monge-Ampère equation Ω := 1/2 A Ω, det D 2 u = f (x), λ f Λ, wit f (x) := f( 1/2 A 1 x). Te section at eigt 1 for u centered at te origin satisfies S 1 (u ) = 1/2 A S,

9 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 9 and by te localization teorem we obtain We remark tat since we obtain B k Ω S 1 (u ) B + k 1. tr U = 1 tr(t U T t ) 1 T 2 tr U, (4.7) g / tr U L C 1/2 log 2 g/ tr U L. In te next lemma we investigate te properties of te rescaled function u. We recall tat if x 0 Ω B ρ ten Ω as an interior tangent ball of radius ρ at x 0, and u satisfies (4.8) ρ x x 0 2 u(x) u(x 0 ) u(x 0 )(x x 0 ) ρ 1 x x 0 2, x Ω. Lemma 4.2. If c, ten a) Ω B 2/k is a grap in te e n direction wose C 1,1 norm is bounded by C 1/2 ; b) for any x, x 0 Ω B 2/k we ave ρ (4.9) 4 x x 0 2 u (x) u (x 0 ) u (x 0 )(x x 0 ) 4ρ 1 x x 0 2, c) if r c small, we ave Proof. For x, x 0 Ω B 2/k we denote ence First we sow tat (4.10) wic is equivalent to u Cr log r 2 in Ω B r. X = T x, X 0 = T x 0, T := 1/2 A 1, x x 0 2 X, X 0 Ω B C 1/2 log. X X 0 1/2 2 x x 0, 1/2 A Z / Z 2, Z := X X 0. Since Ω is C 1,1 in a neigborood of te origin we find ence, if is small Z n C 1/2 log Z A Z Z = τ Z n C 1/2 log 2 Z Z /2, and (4.10) is proved. Part b) follows now from (4.8) and te equality u (x) u (x 0 ) u (x 0 )(x x 0 ) = 1 (u(x) u(x 0) u(x 0 )(X X 0 ).

10 10 N. Q. LE AND O. SAVIN Next we sow tat Ω as small C 1,1 norm. Since Ω as an interior tangent ball at X 0 we see tat (X X 0 ) ν 0 C X X 0 2, were ν 0 is te exterior normal to Ω at X 0. Tis implies, in view of (4.10) (x x 0 ) T t ν 0 C x x 0 2, or (x x 0 ) ν 0 C T t ν 0 x x 0 2, were ν 0 := T t ν 0 / T t ν 0. From te formula for A we see tat ence Since we obtain tus e n ((A 1 )T e n ) = (A 1 e n) e n = 1, (A 1 )T e n 1. ν 0 + e n C 1/2 log (A 1 )T ν 0 1 C 1/2 log A 1 1/2, T t ν 0 = 1/2 (A 1 )T ν 0 1/2 /2. In conclusion (x x 0 ) ν 0 C 1/2 x x 0 2, wic easily implies our claim about te C 1,1 norm of Ω. Next we prove property c). From a), b) above we see tat u satisfies in S 1 (u ) te ypoteses of te Localization Teorem 3.1 at 0 for a small ρ depending on te given constants. We consider a point x 0 Ω B r, and by Lemma 4.1, it remains to sow tat u, S 1 (u ) satisfy te ypoteses of te Localization Teorem 3.1 also at x 0. From (4.4) we ave (4.11) u Cr 2 log r 2 in Ω B 2r, wic, by convexity of u gives n u (x 0 ) Cr log r 2. On te oter and, we use part b) at x 0 and 0 (see (4.9)) and obtain u (x 0 ) + u (x 0 ) (x x 0 ) Cr 2 on Ω B r, tus u (x 0 ) x Cr 2 on Ω B r. Since x n 0 on Ω, we see tat if n u (x 0 ) 0 ten, x u (x 0 ) x Cr 2 if x r/2,

11 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 11 wic gives x u (x 0 ) Cr. We obtain te same conclusion similarly if n u (x 0 ) 0. Te upper bounds on n u (x 0 ) and x u (x 0 ) imply tat if x S 1 (u ) B 1/k we ave u (x 0 ) + u (x 0 ) (x x 0 ) + 1/2 Cr 2 + Cr log r 2 + 1/2 < 1, provided tat r is small. Tis sows tat S x0, 1 2 (u ) S 1 (u ) B 1/k. Moreover, since S x0, 1 (u ) = n/2 S 2 X0, (u) 1 2 we obtain a bound for u(x 0 ). Now we can easily conclude from parts a) and b) and te inclusion above tat u satisfies in S 1 (u ) te ypoteses of te Localization Teorem at x 0 for a small ρ. In te next proposition we compare te distance functions under te following transformations of point and domain: x X := T x, Ω Ω = T Ω, T = 1/2 A 1. Proposition 4.3. For x Ω B + k 1, let X = T x Ω. Ten (see notation (4.1)) 1 C 1/2 log 2 1/2 d Ω (X) d Ω (x) 1 + C 1/2 log 2. Proof. Denote by ξ x, ξ X te unit vectors at x, X wic give te perpendicular direction to Ω respectively Ω, and wic point inside te domain. Since Ω is C 1,1 at te origin and X C 1/2 log we find ξ X e n C 1/2 log. Moreover, te C 1,1 bound of Ω from Lemma 4.2 sows tat ξ x e n C 1/2. We compare 1/2 d Ω (X) wit d Ω (x) by computing te directional derivative of 1/2 d Ω (X) along ξ x. We ave x ( 1/2 d Ω (X)) ξ x = 1/2 X d Ω (X) T ξ x = 1/2 ξ X (T ξ x ) = ξ X (A 1 ξ x) From te inequalities above on ξ x, ξ X we find Using ξ X (A 1 ξ x) e n (A 1 e n) C 1/2 log A 1 C1/2 log 2. e n (A 1 e n) = 1

12 12 N. Q. LE AND O. SAVIN we obtain wic implies our result. x ( 1/2 d Ω (X)) ξ x 1 C 1/2 log 2, 5. Te class D σ and its main properties In tis section we introduce te class D σ tat captures te properties of te rescaled functions u in S 1 (u ). By abuse of notation we use u and Ω wen we define D σ. Fix ρ, λ, Λ. We introduce te class D σ consisting of pairs of function u and domain Ω satisfying te following conditions: (i) 0 Ω, Ω B + 1/k, Ω c 0, (ii) u : Ω R is convex, continuous satisfying u(0) = 0, u(0) = 0, λ det D 2 u Λ; (iii) Ω {u < 1} G {x n σ} were G is a grap in te e n direction wic is defined in B 2/k, and its C 1,1 norm is bounded by σ. (iv) ρ 4 x x 0 2 u(x) u(x 0 ) u(x 0 )(x x 0 ) 4 ρ x x 0 2 x, x 0 G Ω; (v) If r c 0, u C 0 r log r 2 in Ω B r. Te constants k, c 0, C 0 above depend explicitly on ρ, λ, Λ, and n. We remark tat te properties above imply tat if x 0 Ω is close to te origin ten S x0, 1 2 (u) {u < 1}, and u satisfies in S x0, 1 2 (u) te ypoteses of te Localization Teorem at x 0 for some ρ depending on te given constants. Lemma 4.2 can be restated in te following way. Lemma 5.1. Let (u, Ω) be as in Teorem 2.1. Ten, if c, We first construct a useful subsolution. (u, S 1 (u )) D σ wit σ = C 1/2. Lemma 5.2 (Subsolution). Suppose (u, Ω) D δ. If δ c ten te function w := x n u + δ 1 n 1 x 2 + Λn λ n 1 δ x2 n

13 satisfies BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 13 L u (w) := U ij w ij δ 1 n 1 tr U, and on te boundary of te domain D := {x n 2δ} Ω we ave were w 0 on D\F δ, w 1 on F δ, (5.1) F δ := {x n = 2δ, x δ Proof. Let Ten Using te matrix inequality we get p(x) = (n 1) }. ( ) δ 1 n 1 x 2 + Λn λ n 1 δ x2 n. det D 2 p(x) = Λn λ n 1. tr(ab) n(det A det B) 1/n for A, B symmetric 0, L u p = U ij p ij n(det(u) det D 2 p) 1/n = n((det D 2 u) n 1 λ n 1 )1/n nλ. Since δ is small D 2 p δ 1 n 1 I, ence Using L u x n = 0 and we find L u p = U ij p ij δ 1 n 1 tr U. L u u = U ij u ij = n det D 2 u nλ L u w = L u (x n u + 2p) δ 1 n 1 tr U. Next we ceck te beavior of w on D. We decompose D G E δ F δ were E δ := D { x δ On G Ω, we use te properties of D δ and obtain 1 6(n 1) }. u (ρ/4) x 2, x n δ x 2 wic follows from te C 1,1 bound on te grap G. Ten provided tat δ is small. On E δ we use (4.3) and find w (δ + δ 1 n 1 + Cδ) x 2 (ρ/4) x 2 0, 1 u (δ 6(n 1) ) 3 = δ 1 2(n 1) Λn

14 14 N. Q. LE AND O. SAVIN ence, for small δ, w Cδ δ 2(n 1) + cδ n 1 δ 2(n 1) /2 < 0. On F δ, te positive terms in w are bounded by 1/3 for small δ and we obtain w 1. Remark 5.3. For any point x 0 Ω close to te origin we can construct te corresponding subsolution w x0 = z n u x0 + 2p(z) were u x0 := u u(x 0 ) u(x 0 ) (x x 0 ) and wit z denoting te coordinates of te point x in a system of coordinates centered at x 0 wit te z n -axis perpendicular to Ω. From te proof above we see tat w x0 satisfies te same conclusion of Lemma 5.2 if x 0 δ. Next we sow tat u as uniform modulus of convexity on te set F δ introduced above (see (5.1)). Lemma 5.4. Let (u, Ω) D δ. If δ c ten for any y F δ we ave S y,cδ 2(u) Ω. Remark 5.5. From now on we fix te value of δ to be small, universal so tat it satisfies te ypoteses of Lemma 5.2 and Lemma 5.4. Remark 5.6. Since te section S y,cδ 2 /2 is contained in Ω B 1/k and as volume bounded from below we can conclude tat it contains a ball B δ(y) for some δ δ small, universal. We sketc te proof of Lemma 5.4 below. Proof. Let 0 be te maximal value of for wic S y, Ω, and let x 0 S y,0 Ω. Since S y,0 is balanced around y and u grows quadratically away from 0 on G we see tat te point x 0 lies also in a neigborood of te origin. Now we can apply Proposition 3.2 at x 0 and obtain 0 cd Ω (y) 2 cδ 2. A consequence of Lemma 5.2 is te following proposition. Proposition 5.7. Assume (u, Ω) D δ, and let v 0 be a nonnegative function satisfying Ten, for some small θ universal. L u v δ 1 n 1 tr U in Ω, v 1 in Fδ. v 1 2 d G in S θ.

15 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 15 Proof. Lemma 5.2 and te maximum principle for te operator L u imply v w in D, wic gives v(0, x n ) 1 2 x n for x n [0, c]. Te same argument can be repeated at points x 0 Ω if x 0 is sufficiently small, by comparing u wit te corresponding subsolution w x0. We obtain v 1 2 d G in Ω B c, and te lemma follows by coosing θ sufficiently small. Proposition 5.8. Let (u, Ω) D σ, (σ δ) and suppose v satisfies in Ω L u v = g, a d G v b d G, for some a, b [ 1, 1]. Tere exists c 1 small, universal suc tat if ten for some a, b tat satisfy max{σ, g/tr U L } c 1 (b a), a d G v b d G in S θ, a a b b, wit η (0, 1), universal, close to 1. Proof. We define te functions b a η(b a), v 1 = v a d G b a, v 2 = b d G v b a wic are nonnegative. Since v 1 + v 2 = d G, we migt assume (see Remark 5.6) tat te function v 1 satisfies {v 1 δ 2 } B δ(2δe n ) 1 2 B δ(2δe n ). Next we apply Teorem 3.3, for te function Notice tat ṽ 1 v 1 0 in Ω and ṽ 1 := v 1 + c 1 (k 2 x 2 ). L u ṽ 1 (g + σ tr U)(b a) 1 2c 1 tru 0. Using Lemma 5.4 we can apply weak Harnack inequality Teorem 3.3 a finite number of times and obtain ṽ 1 2c 2 > 0 on F δ, for some universal c 2. By coosing c 1 sufficiently small we find v 1 c 2 on F δ.

16 16 N. Q. LE AND O. SAVIN Now we can apply Proposition 5.7 to v 1 /c 2 since provided tat c 1 is small. We obtain ence L u (v 1 /c 2 ) 2(c 1 /c 2 ) tru δ 1 n 1 tr U, v a d G, v 1 (c 2 /2) d G in S θ, a = a + c 2 (b a)/2. 6. Proof of Teorem 2.1 Trougout tis section we assume tat u, v satisfy te ypoteses of Teorem 2.1 and we also assume for simplicity tat u(0) = 0, u(0) = 0. Our boundary gradient estimate states as follows. Proposition 6.1. Let v be as in Teorem 2.1. Ten, in Ω B ρ/2, we ave v(x) C( v L (Ω B ρ) + g/ tr U L (Ω B ρ))d Ω (x). Te proposition follows easily from te construction of a suitable supersolution. Lemma 6.2 (Supersolution). Tere exists universal constants M large, and δ small suc tat te function satisfies and w := Mx n + u δ x 2 L u (w) δ tr U, Λn (λ δ) n 1 x2 n w 0 on (Ω B ρ ), w δ on (Ω B ρ ) \ B ρ/2. Proof. We first coose δ ρ small suc tat u δ x 2 δ on Ω \ B ρ/2. Te existence of δ follows for example from (4.3). We coose M suc tat Ten on Ω, Mx n Λn (λ δ) n 1 x2 n 0 on Ω. w u δ x 2 0, and we obtain te desired inequalities for w on Ω.

17 ten If we denote BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 17 and we obtain as in Lemma 5.2 q(x) := 1 2 det D 2 q = ( δ x 2 + Λn Λn λ n 1, L u w δ tr U. (λ δ) n 1 x2 n D2 q δi Proof of Proposition 6.1. By dividing te equation by a suitable constant we may suppose tat v L δ, g/ tr U L δ, and we need to sow tat v Cd Ω in Ω B ρ/2. Since v w on (Ω B ρ ) and L u v L u w we obtain v w in Ω B ρ ence v(0, x n ) Cx n, ), if x n [0, ρ/2]. Te same argument applies at all points x 0 Ω B ρ/2 and we obtain te upper bound for v. Te lower bound follows similarly and te proposition is proved. Proof of Teorem 2.1. By dividing by a suitable constant we may suppose tat v L + g/ tr U L is sufficiently small suc tat, by Proposition 6.1, v 1 2 d Ω in Ω B ρ/2. We focus our attention on te sections at te origin and we sow tat we can improve tese bounds in te form (6.1) a d Ω v b d Ω, in S, for appropriate constants a, b. First we fix 0 small universal and let Ten we sow by induction tat for all a 0 = 1/2, b 0 = 1/2. = 0 θ k, k 0, we can find a increasing and b decreasing wit k suc tat (6.1) olds and ( ) k 1 + η (6.2) b a = C 1 1/2 log 2. 2 for some large universal constant C 1. We notice tat tis statement olds for k = 0 if 0 is cosen sufficiently small.

18 18 N. Q. LE AND O. SAVIN Assume te statement olds for k. Proposition 4.3 implies tat a d Ω v b d Ω in S 1 (u ) wit a a C 1/2 log 2, b b C 1/2 log 2. Since (u, S 1 (u )) D σ, for σ = C 1/2, and (see (4.6), (4.7)) L u v = g, we can apply Proposition 5.8 and conclude wit g /tru L C 1/2 log 2 g/tru L C 1/2 log 2 c 1 (b a ), a θ d Ω v (x) b θ d Ω, in S θ (u ). b θ a θ η(b a ). Rescaling back to S θ, and using Proposition 4.3 again, we obtain (6.3) a θ d Ω v b θ d Ω, in S θ (u) were b θ a θ η(b a ) + C 1/2 log (1 + η)/2(b a ). By possibly modifying teir values we may take a θ, b θ suc tat From (6.1), (6.2) we find a a θ b θ b, b θ a θ = 1 + η 2 (b a ). osc S v C 1/2+α, for some small α universal. Using (4.2) we obtain osc Br v Cr 1+α if r c, and te teorem is proved.

19 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS Proof of Teorem 2.5 In tis last section we prove Propositions 2.6 and 3.2 and Teorem 2.5. Proof of Proposition 2.6. Let y Ω wit r := d Ω (y) c, and consider te maximal section S,y centered at y, i.e., = max{ By Proposition 3.2 applied at te point we ave and S,y is equivalent to an ellipsoid E i.e were We denote Te rescaling ũ : S 1 R of u S y, Ω}. x 0 S y, Ω, 1/2 r, u(y) u(x 0 ) C 1/2, cẽ S,y y CẼ, E := 1/2 A 1 B 1, wit A, A 1 C log. u y := u u(y) u(y)(x y). ũ( x) := 1 u y (T x) x = T x := y + 1/2 A 1 x, satisfies and det D 2 ũ( x) = f( x) := f(t x), B c S 1 B C, S1 = 1/2 A (S,y y), were S 1 represents te section of ũ at te origin at eigt 1. Te interior C 1,γ estimate for solutions of te Monge-Ampere equation (see [C1]) gives ũ( z 1 ) ũ( z 2 ) C z 1 z 2 γ z 1, z 2 S 1/2 for some γ (0, 1), C universal. Rescaling back and using ũ( z 1 ) ũ( z 2 ) = 1/2 (A 1 )T ( u(z 1 ) u(z 2 )), z 1 z 2 = 1/2 A (z 1 z 2 ) we find u(z 1 ) u(z 2 ) z 1 z 2 γ z 1, z 2 S /2,y. Notice tat tis inequality olds also in te Euclidean ball B r 2(y) S /2,y. Also, if y 0 Ω denotes te closest point to y on Ω i.e y y 0 = r, by Lemma 4.1, we find u(y) u(y 0 ) u(y) u(x 0 ) + u(x 0 ) u(y 0 ) r 1/2.

20 20 N. Q. LE AND O. SAVIN Tese oscillation properties for u and Lemma 4.1 easily imply tat for some α (0, 1), C universal. If we assume tat f C β (Ω) ten [ u] C α ( Ω) C, f C β ( S 1 ) f C β (Ω), and te interior C 2,β estimates for ũ in S 1 (see [C2]) give (7.1) D 2 ũ C β ( S 1/2 ) K. In particular D 2 u(y) = A T D 2 ũ(0)a K log 2 K log r 2, were by K we denote various constants depending on β, f C β (Ω) and te universal constants. Proof of Teorem 2.5. We use te same notations as in te proof of Proposition 2.6. After multiplying v by a suitable constant we may assume tat We define also te rescaling ṽ for v From Teorem 2.4 we obtain for some universal α (0, 1) and C, ence ϕ C 1,1 + g/ tr U L = 1. ṽ(x) := 1/2 v(t x). max v v(x 0 ) v(x 0 )(x x 0 ) C r 1+α S,y max S 1 ṽ( x) ṽ( x 0 ) ṽ( x 0 )( x x 0 ) Cr 1+α 1/2 Cr α. Using te computations in Section 4 we see tat ṽ solves wit Ũ ij ṽ ij = g(x) := 1/2 g(t x), g(x)/tr Ũ L ( S 1/2 ) C1/2 log 2 g/ tr U L Cr α. Since (7.1) olds, we can apply Scauder estimates and find tat for any z 1, z 2 S 1/4 ṽ(0) ṽ( x 0 ) Kr α, ṽ( z 1 ) ṽ( z 2 ) Kr α z 1 z 2 α /2. Using tat ṽ( z i ) = (A 1 )T v(z i ) we obtain v(y) v(x 0 ) Kr α /2, v(z 1 ) v(z 2 ) K z 1 z 2 α /2 for any z 1, z 2 B r 2(y). Tese inequalities and Teorem 2.4 give as in te proof of Proposition 2.6 above te desired C 1,α bound for v.

21 BOUNDARY REGULARITY FOR LINEARIZED MONGE-AMPÈRE EQUATIONS 21 Remark 7.1. Teorem 2.5 still olds if we only assume tat f C(Ω). In tis case one needs to apply te interior C 1,α estimates for te linearized Monge-Ampere equation obtained by Gutiérrez and Nguyen [GN]. We conclude te paper wit a sketc of te proof of Proposition 3.2. Proof of Proposition 3.2. Assume tat te ypoteses of te Localization Teorem 3.1 old at te origin. For a 0 we denote S a := {x Ω u(x) < ax n }, and clearly S a 1 S a 2 if a 1 a 2. Te proposition easily follows once we sow tat S c 1/2 as te sape of te ellipsoid E for all small. From Teorem 3.1 we know S := {u < } k 1 E {x n k 1 1/2 } and since u(0) = 0 we use te convexity of u and obtain (7.2) S k 1/2 S Ω. Tis inclusion sows tat in order to prove tat S is equivalent to E k 1/2 it suffices to bound its volume by below S k c E 1/2. From Teorem 3.1, tere exists y S θ suc tat y n k(θ) 1/2. We evaluate at y and find ũ := u k 1/2 x n, ũ(y) θ k 1/2 k(θ) 1/2 δ, for some δ > 0 provided tat we coose θ small depending on k. Since ũ = 0 on S k 1/2 and det D 2 ũ λ we ave inf ũ C(λ) S k 1/2 2/n, ence c n/2 S k 1/2. References [B] Brenier, Y. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Mat., 44 (1991), no. 4, [C1] Caffarelli, L. A localization property of viscosity solutions to te Monge-Ampère equation and teir strict convexity, Ann. of Mat. 131 (1990), [C2] Caffarelli, L. Interior W 2,p estimates for solutions of Monge-Ampère equation, Ann. of Mat. 131 (1990), [C3] Caffarelli, L. Some regularity properties of solutions of Monge-Ampère equation. Comm. Pure Appl. Mat., 44 (1991), no. 8-9,

22 22 N. Q. LE AND O. SAVIN [CG] Caffarelli, L. A.; Gutiérrez, C. E. Properties of te solutions of te linearized Monge-Ampère equation. Amer. J. Mat. 119 (1997), no. 2, [CNP] Cullen, M. J. P.; Norbury, J.; Purser, R. J. Generalized Lagrangian solutions for atmosperic and oceanic flows. SIAM J. Appl. Anal., 51 (1991), no. 1, [D1] Donaldson, S. K. Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, [D2] Donaldson, S. K. Interior estimates for solutions of Abreu s equation. Collect. Mat. 56 (2005), no. 2, [D3] Donaldson, S. K. Extremal metrics on toric surfaces: a continuity metod. J. Differential Geom. 79 (2008), no. 3, [D4] Donaldson, S. K. Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal. 19 (2009), no. 1, [GN] Gutiérrez, C.; Nguyen, T. Interior gradient estimates for solutions to te linearized Monge-Ampère equations, Preprint. [K] Krylov, N. V. Boundedly inomogeneous elliptic and parabolic equations in a domain. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, [LS2] Le, N. Q.; Savin, O. Some minimization problems in te class of convex functions wit prescribed [Loe] determinant, Preprint on arxiv. Loeper, G. A fully nonlinear version of te incompressible euler equations: te semigeostropic system. SIAM J. Mat. Anal., 38 (2006), no. 3, [S] Savin, O. A localization property at te boundary for te Monge-Ampère equation. arxiv: v2 [mat.ap]. [S2] Savin, O. Pointwise C 2,α estimates at te boundary for te Monge-Ampère equation. arxiv: v1 [mat.ap] [S3] [TW] Savin, O. Global W 2,p estimates for te Monge-Ampère equation. arxiv: v1 [mat.ap]. Trudinger, N. S.; Wang, X. J. Te Bernstein problem for affine maximal ypersurfaces. Invent. Mat. 140 (2000), no. 2, [TW1] Trudinger, N.S. and Wang, X.J., Te affine plateau problem, J. Amer. Mat. Soc. 18(2005), [TW2] Trudinger N.S., Wang X.J, Boundary regularity for Monge-Ampère and affine maximal surface equations, Ann. of Mat. 167 (2008), [TW3] Trudinger, N. S.; Wang, X. J. Te Monge-Ampère equation and its geometric applications. Handbook of geometric analysis. No. 1, , Adv. Lect. Mat. (ALM), 7, Int. Press, Somerville, MA, Department of Matematics, Columbia University, New York, NY address: namle@mat.columbia.edu Department of Matematics, Columbia University, New York, NY address: savin@mat.columbia.edu