Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampere equation

We obtain pointwise $C^{2,\alpha}$ estimates at boundary points for solutions to the Monge-Ampere equation under appropriate local conditions on the right hand side and boundary data.


Introduction
Boundary estimates for solutions of the Dirichlet problem for the Monge-Ampere equation det D 2 u = f in Ω, u = ϕ on ∂Ω, were obtained in the classical paper by Caffarelli, Nirenberg and Spruck [CNS] in the case when ∂Ω, ϕ and f are sufficiently smooth. When f is less regular, i.e f ∈ C α , the global C 2,α estimates were obtained by Trudinger and Wang [TW] for ϕ, ∂Ω ∈ C 3 . In this paper we discuss pointwise C 2,α estimates at boundary points under appropriate local conditions on the right hand side and boundary data. Our main result can be viewed as a natural extension up to the boundary of the pointwise interior C 2,α estimate of Caffarelli in [C2].
We start with the following definition (see [CC]). Definition: Let 0 < α ≤ 1. We say that a function u is pointwise C 2,α at x 0 and write u ∈ C 2,α (x 0 ) if there exists a quadratic polynomial P x0 such that We say that u ∈ C 2 (x 0 ) if Similarly one can define the notion for a function to be C k and C k,α at a point for any integer k ≥ 0.
It is easy to check that if u is pointwise C 2,α at all points of a Lipschitz domain Ω and the equality in the definition above is uniform in x 0 then u ∈ C 2,α (Ω) in the classical sense. Precisely, if there exists M and δ such that for all points x 0 ∈Ω [D 2 u] C α (Ω) ≤ C(δ, Ω)M.
The author was partially supported by NSF grant 0701037.

O. SAVIN
Caffarelli showed in [C2] that if u is a strictly convex solution of det D 2 u = f and f ∈ C α (x 0 ), f (x 0 ) > 0 at some interior point x 0 ∈ Ω, then u ∈ C 2,α (x 0 ). Our main theorem deals with the case when x 0 ∈ ∂Ω.
Theorem 1.1. Let Ω be a convex domain and let u :Ω → R convex, continuous, solve the Dirichlet problem for the Monge-Ampere equation with positive, bounded right hand side i.e for some constants λ, Λ.
The way ϕ separates locally from the tangent plane at x 0 is given by the tangential second derivatives of u at x 0 . Thus the assumption that this separation is quadratic is in fact necessary for the C 2,α estimate to hold. Heuristically, Theorem 1.1 states that if the tangential pure second derivatives of u are bounded below then the boundary Schauder estimates hold for the Monge-Ampere equation.
A more precise, quantitative version of Theorem 1.1 is given in section 7 (see Theorem 7.1).
Given the boundary data, it is not always easy to check the quadratic separation since it involves some information about the slope of the tangent plane at x 0 . However, this can be done in several cases (see Proposition 3.2). One example is when ∂Ω is uniformly convex and ϕ, ∂Ω ∈ C 3 (x 0 ). The C 3 condition of the data is optimal as it was shown by Wang in [W]. Other examples are when ∂Ω is uniformly convex and ϕ is linear, or when ∂Ω is tangent of second order to a plane at x 0 and ϕ has quadratic growth near x 0 .
As a consequence of Theorem 1.1 we obtain a pointwise C 2,α estimate in the case when the boundary data and the domain are pointwise C 3 . As mentioned above, the global version was obtained by Trudinger and Wang in [TW].
We also obtain the C 2,α estimate in the simple situation when ∂Ω ∈ C 2,α and ϕ is constant.
The key step in the proof of Theorem 1.1 is a localization theorem for boundary points which was recently proved in [S]. It states that under natural local assumptions on the domain and boundary data, the sections with x 0 ∈ ∂Ω are "equivalent" to ellipsoids centered at x 0 .
If ϕ separates quadratically from the tangent plane of u at x 0 , then for each small h > 0 there exists an ellipsoid E h of volume h n/2 such that with c, C constants independent of h.
Theorem 1.4 is an extension up to the boundary of the localization theorem at interior points due to Caffarelli in [C1]. For completeness we provide also its proof in the current paper.
The paper is organized as follows. In section 2 we discuss briefly the compactness of solutions to the Monge-Ampere equation which we use later in the paper (see Theorem 2.7). For this we need to consider also solutions with possible discontinuities at the boundary. In section 3 we give a quantitative version of the Localization Theorem (see Theorem 3.1). In sections 4 and 5 we provide the proof of Theorem 3.1. In section 6 we obtain a version of the classical Pogorelov estimate in half-domain (Theorem 6.4). Finally, in section 7 we use the previous results together with a standard approximation method and prove our main theorem.

Solutions with discontinuities on the boundary
Let u : Ω → R be a convex function with Ω ⊂ R n bounded and convex. Denote by Definition 2.1. We define the values of u on ∂Ω to be equal to ϕ i.e From the definition we see that ϕ is lower semicontinuous.
with f ≥ 0 continuous and bounded on Ω, then there exists an increasing sequence of subsolutions, continuous up to the boundary, with lim u n = u inΩ, where the values of u on ∂Ω are defined as above.
Indeed, let us assume for simplicity that 0 ∈ Ω, u(0) = 0, u ≥ 0. Then, on each ray from the origin u is increasing, is an increasing family of continuous functions as ε → 0, with In order to obtain a sequence of subsolutions we modify v ε as The claim is proved since as ε → 0 we can choose w ε to converge uniformly to 0.
Proof. Since u can be approximated by a sequence of continuous functions onΩ it suffices to prove the result in the case when u is continuous onΩ and u < v on ∂Ω. Then, u < v in a small neighborhood of ∂Ω and the inequality follows from the standard comparison principle.
A consequence of the comparison principle is that a solution det D 2 u = f is determined uniquely by its boundary values u| ∂Ω .
Next we define the notion of convergence for functions which are defined on different domains. Definition 2.3. a) Let u k : Ω k → R be a sequence of convex functions with Ω k convex. We say that u k converges to u : Ω → R i.e u k → u if the upper graphs convergē U k →Ū in the Haudorff distance.
In particular it follows thatΩ k →Ω in the Hausdorff distance. b) Let ϕ k : ∂Ω k → R ∪ {∞} be a sequence of lower semicontinuous functions. We say that ϕ k converges to ϕ : f k → f uniformly on compact sets of Ω.
Remark: When we restrict the Hausdorff distance to the nonempty closed sets of a compact set we obtain a compact metric space. Thus, if Ω k , u k are uniformly bounded then we can always extract a convergent subsequence u km → u. Similarly, if Ω k , ϕ k are uniformly bounded we can extract a convergent subsequence ϕ km → ϕ.
Proposition 2.4. Let u k :Ω k → R be continuous and where ϕ * is the convex envelope of ϕ on ∂Ω i.e Φ * is the restriction to ∂Ω × R of the convex hull generated by Φ.
Indeed consider a hyperplane which lies below K. Then u k − l ≥ 0 on ∂Ω k and by Alexandrov estimate we have that u k − l ≥ −Cd 1/n k where d k represents the distance to ∂Ω k . By taking k → ∞ we see that u − l ≥ −Cd 1/n thus no point on ∂Ω × R below the hyperplane belongs toŪ .
Proposition 2.4 says that given any ϕ bounded and lower semicontinuous, and f ≥ 0 bounded and continuous we can always solve uniquely the Dirichlet problem det D 2 u = f in Ω, u = ϕ on ∂Ω by approximation. Indeed, we can find sequences ϕ k , f k of continuous, uniformly bounded functions defined on strictly convex domains Ω k such that ϕ k → ϕ and f k → f . Then the corresponding solutions u k are uniformly bounded and continuous up to the boundary. Using compactness and the proposition above we see that u k must converge to the unique solution u in (2.1).
We extend the Definition 2.1 in order to allow a boundary data that is not necessarily convex.
Definition 2.5. Let ϕ : ∂Ω → R be a lower semicontinuous function. When we write that a convex function u satisfies where ϕ * is the convex envelope of ϕ on ∂Ω.
Whenever ϕ * and ϕ do not coincide we can think of the graph of u as having a vertical part on ∂Ω between ϕ * and ϕ.
It follows easily from the definition above that the boundary values of u when we restrict to the domain The advantage of introducing the notation of Definition 2.5 is that the boundary data is preserved under limits.
Proposition 2.6. Assume with Ω k , ϕ k uniformly bounded and Proof. Using the compactness of solutions we may assume that u k converges to a limit u and it remains to prove that u = ϕ on ∂Ω.
Denote byũ k the restriction of u k to the set Notice that for fixed k, as ε k → 0 thenũ k → u k andφ k → ϕ * k . On the other hand, from the hypotheses we obtain that ϕ * k → ϕ * . Thus, we can choose a sequence of ε k → 0 such thatũ k → u,φ k → ϕ * ,f k → f. Now, sinceũ k are continuous up to the boundary, the conclusion follows from Proposition 2.4.
Finally, we state a version of the last proposition for solutions with bounded right-hand side i.e λ ≤ det D 2 u ≤ Λ, where the two inequalities are understood in the viscosity sense.
and Ω k , ϕ k uniformly bounded. Then there exists a subsequence k m such that

The Localization Theorem
In this section we state the quantitative version of the localization theorem at boundary points (Theorem 3.1).
Let u : Ω → R be continuous, convex, satisfying We extend u to be ∞ outside Ω. After subtracting a linear function we assume that (3.3) x n+1 = 0 is the tangent plane to u at 0, in the sense that u ≥ 0, u(0) = 0, and any hyperplane x n+1 = εx n , ε > 0, is not a supporting plane for u. We investigate the geometry of the sections of u at 0 that we denote for simplicity of notation S h := {x ∈ Ω : u(x) < h}.
We show that if the boundary data has quadratic growth near {x n = 0} then, as h → 0, S h is equivalent to a half-ellipsoid centered at 0.
Precisely, our theorem reads as follows.
Theorem 3.1 (Localization Theorem). Assume that Ω, u satisfy (3.1)-(3.3) above and for some µ > 0, Then, for each h < c(ρ) there exists an ellipsoid E h of volume h n/2 such that Moreover, the ellipsoid E h is obtained from the ball of radius h 1/2 by a linear transformation A −1 h (sliding along the x n = 0 plane) The constant k above depends on µ, λ, Λ, n and c(ρ) depends also on ρ.
The ellipsoid E h , or equivalently the linear map A h , provides information about the behavior of the second derivatives near the origin. Heuristically, the theorem states that in S h the tangential second derivatives are bounded from above and below and the mixed second derivatives are bounded by | log h|.
The hypothesis that u is continuous up to the boundary is not necessary, we just need to require that (3.4) holds in the sense of Definition 2.5.
Given only the boundary data ϕ of u on ∂Ω, it is not always easy to check the main assumption (3.4) i.e that ϕ separates quadratically on ∂Ω (in a neighborhood of {x n = 0}) from the tangent plane at 0. Proposition 3.2 provides some examples when this is satisfied depending on the local behavior of ∂Ω and ϕ (see also the remarks below). 1) ϕ is linear in a neighborhood of 0 and Ω is uniformly convex at the origin.
2) ∂Ω is tangent of order 2 to {x n = 0} and ϕ has quadratic growth in a neighborhood of {x n = 0}.
Proof. 1) Assume ϕ = 0 in a neighborhood of 0. By the use of standard barriers, the assumptions on Ω imply that the tangent plane at the origin is given by for some bounded µ > 0. Then (3.4) clearly holds.
2) After subtracting a linear function we may assume that on ∂Ω in a neighborhood of {x n = 0}. Using a barrier we obtain that l 0 , the tangent plane at the origin, has bounded slope. But ∂Ω is tangent of order 2 to {x n = 0}, thus l 0 grows less than quadratic on ∂Ω in a neighborhood of {x n = 0} and (3.4) is again satisfied.
3) Since Ω is uniformly convex at the origin, we can use barriers and obtain that l 0 has bounded slope. After subtracting this linear function we may assume l 0 = 0. Since ϕ, ∂Ω ∈ C 3 (0) we find that with Q 0 a cubic polynomial. Now ϕ ≥ 0, hence Q 0 has no linear part and its quadratic part is given by, say We need to show that µ i > 0. If µ 1 = 0, then the coefficient of x 3 1 is 0 in Q 0 . Thus, if we restrict to ∂Ω in a small neighborhood near the origin, then for all small h the set {ϕ < h} contains |S h | ≤ Ch n/2 , for some C depending on λ and n, and we contradict the inequality above as h → 0.
Remark 3.4. From above we see that we can often verify (3.4) in the case when ϕ, ∂Ω ∈ C 1,1 (0) and Ω is uniformly convex at 0. Indeed, if l ϕ represents the tangent plane at 0 to ϕ : ∂Ω → R (in the sense of (3.3)), then (3.4) holds if either ϕ separates from l ϕ quadratically near 0, or if ϕ is tangent to l ϕ of order 3 in some tangential direction.

Proof of Theorem 3.1 (I)
We prove Theorem 3.1 in the next two sections. In this section we obtain some preliminary estimates and reduce the theorem to a statement about the rescalings of u. This statement is proved in section 5 using compactness.
Next proposition was proved by Trudinger and Wang in [TW]. It states that the volume of S h is proportional to h n/2 and after an affine transformation (of controlled norm) we may assume that the center of mass of S h lies on the x n axis. Since our setting is slightly different we provide its proof.
such that the rescaled functionũ The constant k 0 above depends on µ, λ, Λ, n and the constants C(ρ), c(ρ) depend also on ρ.
In this section we denote by c, C positive constants that depend on n, µ, λ, Λ. For simplicity of notation, their values may change from line to line whenever there is no possibility of confusion. Constants that depend also on ρ are denote by c(ρ), C(ρ).

Proof. The function
Otherwise, from (4.1) and John's lemma we obtain for some large C 1 = C 1 (ρ). Then the function and for all small h, and det D 2 w = 2Λ. Hence w ≤ u in S h , and we contradict that 0 is the tangent plane at 0. Thus claim (4.2) is proved. Now, define

The center of mass ofS
and lies on the x n -axis from the definition of A h . Moreover, since x * h ∈ S h , we see from (4.1)-(4.2) that and this proves (i).
Next we prove (ii). From John's lemma, we know that after relabeling the x ′ coordinates if necessary, )}, we see that the domain of definition of g h contains a ball of radius (µh/2) 1/2 . This implies that d i ≥ c 1 h 1/2 , i = 1, · · · , n − 1, for some c 1 depending only on n and µ. Also from (4.2) we see that with k 0 small depending only on µ, n, Λ, which gives the left inequality in (ii).
To this aim we consider the barrier, We choose c sufficiently small depending on µ, n, Λ so that for all h < c(ρ), and on the part of the boundaryG h , we have w ≤ũ since Moreover, if our claim does not hold, then thus w ≤ũ inS h . By definition,ũ is obtained from u by a sliding along x n = 0, hence 0 is still the tangent plane ofũ at 0. We reach again a contradiction sincẽ u ≥ w ≥ εx n and the claim is proved. Finally we show that for some C depending only on λ, n.
we obtain the desired conclusion.
In the proof above we showed that for all h ≤ c(ρ), the entries of the diagonal matrix D h from (4.3) satisfy The main step in the proof of Theorem 1.1 is the following lemma that will be completed in Section 5.
Lemma 4.2. There exist constants c, c(ρ) such that Using Lemma 4.2 we can easily finish the proof of our theorem.
Proof of Theorem 1.1. Since all d i are bounded below by ch 1/2 and their product is bounded above by Ch n/2 we see that for all h ≤ c(ρ). Using (4.3) we obtaiñ We define the ellipsoid E h as Comparing the sections at levels h and h/2 we find and we easily obtain the inclusion then the inclusion above implies which gives the desired bound In order to prove Lemma 4.2 we introduce a new quantity b(h) which is proportional to d n h −1/2 and is appropriate when dealing with affine transformations.

Notation. Given a convex function
Whenever there is no possibility of confusion we drop the subindex u and use the notation b(h).
Below we list some basic properties of b(h).
given by a linear transformation A which leaves the x n coordinate invariant does not change the value of b, i.e bũ(h) = b u (h).

3) If
4) If we multiply u by a constant, i.e.
From (4.3) and property 2 above, hence Lemma 4.2 will follow if we show that b(h) is bounded below. We achieve this by proving the following lemma.
This lemma states that if the value of b(h) on a certain section is less than a critical value c 0 , then we can find a lower section at height still comparable to h where the value of b doubled. Clearly Lemma 4.3 and property 1 above imply that b(h) remains bounded for all h small enough.
The quotient in (4.6) is the same forũ which is defined in Proposition 4.1. We normalize the domainS h andũ by considering the rescaling Then ch −1/2 ≤ γ ≤ Ch −1/2 , and the diagonal entries of A satisfy a i ≥ c, i = 1, 2, · · · , n − 1, Also, from Proposition 4.1 on the part G of the boundary of ∂Ω v where {v < 1} we have In order to prove Lemma 4.3 we need to show that if σ, a n are sufficiently small depending on n, µ, λ, Λ then the function v above satisfies for some 1 > t ≥ c 0 .
Since α < 1, the smallness condition on σ is satisfied by taking h < c(ρ) sufficiently small. Also a n being small is equivalent to one of the a i , 1 ≤ i ≤ n − 1 being large since their product is 1 and a i are bounded below.
In the next section we prove property (4.8) above by compactness, by letting σ → 0, a i → ∞ for some i (see Proposition 5.1).

Proof of Theorem 3.1 II
In this section we consider the class of solutions v that satisfy the properties above. After relabeling the constants µ and a i , and by abuse of notation writing u instead of v, we may assume we are in the following situation.
Fix µ small and λ, Λ. For an increasing sequence a 1 ≤ a 2 ≤ . . . ≤ a n−1 with a 1 ≥ µ, we consider the family of solutions u ∈ D µ σ (a 1 , a 2 , . . . , a n−1 ) of convex functions u : Ω → R that satisfy Moreover we assume that the boundary ∂Ω has a closed subset G which is a graph in the e n direction with projection π n (G) ⊂ R n−1 along e n (5.5) and (see Definition 2.5), the boundary values of u = ϕ on ∂Ω satisfy (5.6) ϕ = 1 on ∂Ω \ G; In this section we prove Proposition 5.1. For any M > 0 there exists C * depending on M, µ, λ, Λ, n such that if u ∈ D µ σ (a 1 , a 2 , . . . , a n−1 ) with a n−1 ≥ C Clearly the function v of the previous section satisfies the hypotheses above (after renaming the constant µ) provided that σ, a n are sufficiently small.
Proposition 5.1 follows easily from the next proposition.
Proposition 5.2. For any M > 0 and 0 ≤ k ≤ n − 2 there exists c k depending on M, µ, λ, Λ, n, k such that if Indeed, if Proposition 5.1 fails for a sequence of constants C * → ∞ then we obtain a limiting solution u as in (5.12) for which b(h) ≤ M for all h > 0. This contradicts Proposition 5.2 (with M replaced by 2M ).
We prove Proposition 5.2 by induction on k. We start by introducing some notation. Denote Definition 5.3. We say that a linear transformation T : R n → R n is a sliding along the y direction if We see that T leaves the (z, x n ) components invariant together with the subspace (y, 0, 0). Clearly, if T is a sliding along the y direction then so is T −1 and det T = 1.
The key step in the proof of Proposition 5.2 is the following lemma.
Lemma 5.4. Assume that u ≥ p(|z| − qx n ), for some p, q > 0 and assume that for each section S h of u, h ∈ (0, 1), there exists T h a sliding along the y direction such that . . , 1, ∞, . . . , ∞). Proof. Assume by contradiction that u ∈ D µ 0 and it satisfies the hypotheses with q ≤ q 0 for some q 0 . We show that for some 0 < p ′ ≪ p, where the constant η > 0 depends only on q 0 and µ, C 0 , Λ, n. Then, since q ′ ≤ q 0 , we can apply this result a finite number of times and obtain u ≥ ε(|z| + x n ), for some small ε > 0. This gives S h ⊂ {x n ≤ ε −1 h} hence and by the hypothesis above and we contradict (5.3). Now we prove (5.13). Since u ∈ D µ 0 as above, there exists a closed set such that on the subspace (y, 0, 0) Let w be a rescaling of u, and our hypothesis becomes (5.14) w ≥ p h 1/2 (|z| − qx n ). Moreover the boundary values ϕ w of w on ∂Ω w satisfy Next we show that ϕ w ≥ v on ∂Ω w where v is defined as v := δ|x| 2 + Λ δ n−1 (z 1 − qx n ) 2 + N (z 1 − qx n ) + δx n , and δ is small depending on µ and C 0 , and N is chosen large such that Λ δ n−1 t 2 + N t is increasing in the interval |t| ≤ (1 + q 0 )C 0 .
From the definition of v we see that On the part of the boundary ∂Ω w where z 1 ≤ qx n we use that Ω w ⊂ B C0 and obtain v ≤ δ(|x| 2 + x n ) ≤ ϕ w .
On the part of the boundary ∂Ω w where z 1 > qx n we use (5.14) and obtain with C arbitrarily large provided that h is small enough. We choose C such that the inequality above implies Λ µ n−1 (z 1 − qx n ) 2 + N (z 1 − qx n ) < 1 2 .

O. SAVIN
Then In conclusion ϕ w ≥ v on ∂Ω w hence the function v is a lower barrier for w in Ω w . Then w ≥ N (z 1 − qx n ) + δx n and, since this inequality holds for all directions in the z-plane, we obtain Scaling back we get Since u is convex and u(0) = 0, this inequality holds globally, and (5.13) is proved.

Proof. By compactness we need to show that there does not exist
S h ⊂ Ch 1/2 B + 1 , and we contradict Lemma 5.4 for k = 0.

Now we prove Proposition 5.2 by induction on k.
Proof of Proposition 5.2. In this proof we denote by c, C positive constants that depend on M, µ, λ, Λ, n and k.
We assume that the proposition holds for all nonnegative integers up to k − 1, 1 ≤ k < n − 2, and we prove it for k. Let u ∈ D µ 0 (a 1 , . . . , a k , ∞, . . . , ∞). By the induction hypotheses and compactness we see that there exists a constant C k (µ, M, λ, Λ, n) such that if a k ≥ C k then b(h) ≥ M for some h ≥ C −1 k . Thus, it suffices to consider only the case when a k < C k .
We show that such a function u does not exist. Denote as before in B + 1/μ . As before we obtain that the inequality above holds in Ω, hence From (5.16)-(5.17) we see that the section S h of u satisfies From John's lemma we know that S h is equivalent to an ellipsoid E h of the same volume i.e h the center of mass of S h . For any ellipsoid E h in R n of positive volume we can find T h , a sliding along the y direction (see Definition 5.3), such that (5.20) with a matrix A that leaves the (y, 0, 0) and (0, z, x n ) subspaces invariant, and det A = 1. By choosing an appropriate system of coordinates in the y and z variables we may assume in fact that with 0 < β 1 ≤ · · · ≤ β k , and (5.20) and that 0 ∈ ∂S h , we obtain (5.21)x * h + ch 1/2 AB 1 ⊂S h ⊂ Ch 1/2 AB 1 , det A = 1, for the matrix A as above and withx * h the center of mass ofS h . Next we use the induction hypothesis and show thatS h is equivalent to a ball.
we see that if b u (h) (and therefore θ n ) becomes smaller than a critical value c * then withM := 2μ −1 , and by the induction hypothesis which implies b u (hh) ≥ 2b u (h) and our claim follows. Next we claim that γ j are bounded below by the same argument. Indeed, from the claim above θ n is bounded below and if some γ j is smaller than a small valuẽ c * then β k ≥ C k (μ,M 1 , λ, Λ, n)

By the induction hypothesis
which gives b u (hh) ≥ 2M , contradiction. In conclusion θ n , γ j are bounded below which implies that β i are bounded above. This shows that |A| is bounded and the lemma is proved.

Pogorelov estimate in half-domain
In this section we obtain a version of Pogorelov estimate at the boundary (Theorem 6.4 below). A similar estimate was proved also in [TW]. We start with the following a priori estimate.
Proposition 6.1. Let u :Ω → R, u ∈ C 4 (Ω) satisfy the Monge-Ampere equation Assume that for some constant k > 0, We divide the proof into four steps.
For each x 0 ∈ {|x ′ | ≤ k, x n = 0}, we consider the barrier This gives a lower bound for u n (x 0 ). Moreover, writing the inequality for all x 0 with |x 0 | = k we obtain D ⊂ {x n ≥ c(|x ′ | − k)}. From the values of u on {x n = 0} and the inclusion above we obtain a lower bound on u n on ∂D in a neighborhood of {x n = 0}. Since Ω contains the cone generated by ke n and {|x ′ | ≤ 1, x n = 0} and u ≤ 1 in Ω, we can use the convexity of u and obtain also an upper bound for u n and all |u i |, 1 ≤ i ≤ n − 1, on ∂D in a neighborhood of {x n = 0}. We find where c 0 > 0 is a small constant depending on k and n. We obtain a similar bound on ∂D ∩ {x n ≥ c 0 } by bounding below by a small positive constant. Indeed, if y ∈ ∂Ω ∩ {x n ≥ c 0 /2}, then there exists a linear function l y with bounded gradient so that u(y) = l y (y), u ≥ l y on ∂Ω.
Then, using Alexandrov estimate for (u − l y ) − we obtain hence D stays outside a fixed neighborhood of y.
Step 2: We show that It suffices to prove that |u in | are bounded in E with i = 1, .., n − 1. Let L ϕ := u ij ϕ ij denote the linearized Monge-Ampere operator for u. Then and if we define P (x) = δ|x ′ | 2 + δ 1−n x 2 n then L P = T r (D 2 u) −1 D 2 P ≥ n det(D 2 u) −1 det D 2 P 1 n ≥ n.
, where l x0 denotes the supporting linear function for u at x 0 , δ = 1/4, and γ 1 , and, since u is Lipschitz in D we can choose γ 1 , γ 2 large, depending only on k and n such that v x0 ≤ u i on ∂D.
This shows that the inequality above holds also in D and we obtain a lower bound on u in (x 0 ). Similarly we obtain an upper bound.
Step 3: We show that We apply the classical Pogorelov estimate in the set Precisely if the maximal value of log 1 4 k 2 − u + log u ii + 1 2 u 2 i occurs in the interior of F then this value is bounded by a constant depending only on n and max F |∇u| (see [C2]). From step 2, the expression is bounded above on ∂F and the estimate follows.
Step 4: The Monge-Ampere equation is uniformly elliptic in {u < k 2 /8} and by Evans-Krylov theorem and Schauder estimates we obtain the desired C 3,1 bound.
Remark 6.2. Assume the boundary values of u are given by with p(x ′ ) a quadratic polynomial that satisfies for some ρ > 0. Then u C 3,1 ({u< 1 16 k 2 }) ≤ C(ρ, k, n). Indeed, after an affine transformation we can reduce the problem to the case p(x ′ ) = |x ′ | 2 /2. Remark 6.3. Proposition 6.1 holds as well if we replace the half-space {x n ≥ 0} with a large ball of radius ε −1 the boundary values of u satisfy then for all small ε, u C 3,1 ({u<k 2 /16}) ≤ C, with C depending only on k and n.
The proof is essentially the same except that in the barrier functions w x0 , v x0 we need to replace x n by (x − x 0 ) · ν x0 where ν x0 denotes the inner normal to ∂Ω at x 0 , and in step 2 we work (as in [CNS]) with the tangential derivative As a consequence of the Proposition 6.1 and the remarks above we obtain Assume that for some constants ρ, k > 0,

and (see Definition 2.5) the boundary values of u are given by
where p is a quadratic polynomial that satisfies Then (6.1) u C 3,1 (B + c 0 ) ≤ c −1 0 , with c 0 > 0 small, depending only on k, ρ and n.
Proof. We approximate u on ∂Ω by a sequence of smooth functions u m on ∂Ω m , with Ω m smooth, uniformly convex, so that u m , Ω m satisfy the conditions of Remark 6.3 above. Notice that u m is smooth up to the boundary by the results in [CNS], thus we can use Proposition 6.1 for u m . We let m → ∞ and obtain (6.1) since B + c0 ⊂ {u < k 2 /16}, by convexity.

Pointwise C 2,α estimates at the boundary
Let Ω be a bounded convex set with , for some small ρ > 0, that is Ω ⊂ (R n ) + and Ω contains an interior ball tangent to ∂Ω at 0.
We also assume that on ∂Ω, in a neighborhood of {x n = 0}, u separates quadratically from the tangent plane {x n+1 = 0}, Our main theorem is the following.
A consequence of the proof of Theorem 7.1 is that if f ∈ C α near the origin, then u ∈ C 2,α in any cone C θ of opening θ < π/2 around the x n -axis i.e Corollary 7.2. Assume u satisfies the hypotheses of Theorem 7.1 and Given any θ < π/2 there exists δ(M, θ) small, such that We also mention the global version of Theorem 7.1.
In general, the Lipschitz bound is easily obtained from the boundary data u| ∂Ω . We can always do this if for example Ω is uniformly convex.
The proof of Theorem 7.1 is similar to the proof of the interior C 2,α estimate from [C2], and it has three steps. First we use the localization theorem to show that after a rescaling it suffices to prove the theorem only in the case when M is arbitrarily small (see Lemma 7.4). Then we use Pogorelov estimate in half-domain (Theorem 6.4) and reduce further the problem to the case when u is arbitrarily close to a quadratic polynomial (see Lemma 7.5). In the last step we use a standard iteration argument to show that u is well-approximated by quadratic polynomials at all scales.
We assume for simplicity that f (0) = 1, otherwise we divide u by f (0). Constants depending on ρ, λ, Λ, n and α are called universal. We denote them by C, c and they may change from line to line whenever there is no possibility of confusion. Constants depending on universal constants and other parameters i.e M, σ, δ, etc. are denoted as C(M, σ, δ).
We denote linear functions by l(x) and quadratic polynomials which are homogenous and convex we denote by p(x ′ ), q(x ′ ), P (x).
The localization theorem says that the section S h is comparable to an ellipsoid E h which is obtained from B h 1/2 by a sliding along {x n = 0}. Using an affine transformation we can normalize S h so that it is comparable to B 1 . In the next lemma we show that, if h is sufficiently small, the corresponding rescaling u h satisfies the hypotheses of u in which the constant M is replaced by an arbitrary small constant σ.
Proof. By the localization theorem Theorem 3.1, for all h ≤ c, Then we define u h as above and obtain if h 0 is small hence, since Ω has an interior tangent ball of radius ρ, we have Then |ν h y n | ≤ k −1 | log h||y ′ | 2 ≤ |y ′ |/2, We obtain and also In the next lemma we show that if σ is sufficiently small, then u h can be wellapproximated by a quadratic polynomial near the origin.
Lemma 7.5. For any δ 0 , ε 0 there exist σ 0 (δ 0 , ε 0 ), µ 0 (ε 0 ) such that for any function u h satisfying properties a), b), c) of Lemma 7.4 with σ ≤ σ 0 we can find a rescaling for some P 0 , quadratic polynomial, Proof. We prove the lemma by compactness. Assume by contradiction that the statement is false for a sequence u m satisfying a), b), c) of Lemma 7.4 with σ m → 0. Then, we may assume after passing to a subsequence if necessary that p m → p ∞ , q m → 0 uniformly on B k −1 , and u m : S 1 (u m ) → R converges to (see Definition 2.3) Then, by Theorem 2.6, u ∞ satisfies l ∞ := γ ∞ x n , |γ ∞ | ≤ c −1 0 , and P ∞ is a quadratic polynomial such that Choose µ 0 small such that Then, for all large m,ũ Finally, we let P m be a perturbation of P ∞ such that Thenũ m , P m ,p m ,q m satisfy the conclusion of the lemma for all large m, and we reached a contradiction.
2) on ∂Ω ∩ B 1 we havep,q so that By choosing δ 0 , ε 0 appropriately small, universal, we show in Lemma 7.6 that there existl,P such that with C a universal constant. Rescaling back, we obtain that u is well approximated by a quadratic polynomial at the origin i.e |u − l − P | ≤ C(M )|x| 2+α in Ω ∩ B ρ , and |∇l|, D 2 P ≤ C(M ) which, by (7.3), proves Theorem 7.1. Since α ∈ (0, 1), in order to prove thatũ ∈ C 2,α (0) it suffices to show thatũ is approximated of order 2 + α by quadratic polynomials l m + P m in each ball of radius r m 0 for some small r 0 > 0, and thenl +P is obtained in the limit as m → ∞ (see [C2], [CC]). Thus Theorem 7.1 follows from the next lemma.
From the definition of v and the properties of P m we see that in B 1 |v − P m | ≤ r −2 |(ũ −p)(rx)| + |γ m ||x n /r −q| + 2nC 0 |x n |, and the inequalities above and property 2) imply with C 1 universal constant (depending only on n and C 0 ). We want to compare v with the solution w : B + 1/8 → R, det D 2 w = 1, which has the boundary conditions w = v on ∂B + 1/8 ∩ Ω v w = P m on ∂B + 1/8 \ Ω v . In order to estimate |u − w| we introduce a barrier φ defined as where c(β) is chosen such that φ = 1 on ∂B 1/2 and φ = 0 on ∂B 1/4 . We choose the exponent β > 0 depending only on C 0 and n such that for any symmetric matrix A with (2C 0 ) 1−n I ≤ A ≤ (2C 0 ) n−1 I, we have T r A(D 2 φ) ≤ −η 0 < 0, for some η 0 small, depending only on C 0 and n.
Remark 7.7. The proof of Lemma 7.6 applies also at interior points. More precisely, ifũ satisfies the hypotheses in B 1 (x 0 ) ⊂Ω instead of B 1 ∩Ω then the conclusion holds in B 1 (x 0 ). The proof is in fact simpler since, in this case we take w so that w = v on ∂B 1 (x 0 ), and then (7.9) is automatically satisfied, so there is no need for the barrier φ. Also, at the end we apply the classical interior estimate of Pogorelov instead of the estimate in half-domain. Now we can sketch a proof of Corollary 7.2 and Theorem 7.3. If u satisfies the conclusion of Theorem 7.1 then, after an appropriate dilation, any point in C θ ∩ B δ becomes an interior point x 0 as in Remark 7.7 above for the rescaled functionũ. Moreover, the hypotheses of Lemma 7.6 hold in B 1 (x 0 ) for some appropriate ε ≤ ε 0 . Then Corollary 7.2 follows easily from Remark 7.7.
If u satisfies the hypotheses of Theorem 7.3 then we obtain as above that u C 2,α (D δ ) ≤ C, D δ := {x ∈ Ω| dist(x, ∂Ω) ≤ δ}, for some δ and C depending on the data. We combine this with the interior C 2,α estimate of Caffarelli in [C2] and obtain the desired bound.