Global $W^{2,p}$ estimates for the Monge-Ampere equation

We obtain global $W^{2,p}$ estimates for the Monge-Ampere equation under natural assumptions on the domain and boundary data.


Introduction
Interior W 2,p estimates for strictly convex solutions for the Monge-Ampere equation were obtained by Caffarelli in [C] under the necessary assumption of small oscillation of the right hand side. The theorem can be stated as follows.
Theorem 1.1 (Interior W 2,p estimates). Let u : Ω → R, be a continuous convex solution to the Monge-Ampere equation for some positive constants λ, Λ. For any p, 1 < p < ∞ there exists ε(p) > 0 depending on p and n such that if for some small ρ > 0, then u W 2,p ({u<−ρ}) ≤ C, and C depends on ρ, λ, Λ, p and n.
In this short paper we obtain the global W 2,p estimates under natural assumptions on the domain and boundary data.
Let Ω be a convex bounded domain and u : Ω → R be a Lipschitz continuous convex solution of the Monge-Ampere equation (1.2). Assume that u| ∂Ω , ∂Ω ∈ C 1,1 , and there exists ρ > 0 small such that f satisfies (1.3). If u separates quadratically on ∂Ω from its tangent planes i.e The author was partially supported by NSF grant 0701037.
Remark 1.3. The gradient ∇u(x) when x ∈ ∂Ω is understood in the sense that is a supporting hyperplane for the graph of u(y) but is not a supporting hyperplane for any δ > 0, where ν x denotes the exterior normal to ∂Ω at x.
In general the Lipschitz continuity of the solution can be easily obtained from the boundary data by the use of barriers. Also the quadratic separation assumption (1.4) can be checked in several situations directly from the boundary data, see Proposition 3.2 in [S2]. This is the case for example when the boundary data is more regular i.e (1.5) u| ∂Ω , ∂Ω ∈ C 3 , and Ω is uniformly convex.
The assumptions on the boundary behavior of u and ∂Ω in Theorem 1.2 and Corollary 1.4 seem to be optimal. Wang in [W] gave examples of solutions u to (1.2) with f = 1 and either u ∈ C 2,1 or ∂Ω ∈ C 2,1 , that do not belong to W 2,p (Ω) for large values of p.
The proof of Theorem 1.2 is based on a localization theorem for the Monge-Ampere equation at boundary points which was proved in [S1], [S2]. 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 . We give its precise statement below.
Assume for simplicity that and Ω contains an interior ball tangent to ∂Ω at 0. Let u : Ω → R be continuous, convex, satisfying After subtracting a linear function we also assume that (1.8) 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 = δx n , δ > 0, is not a supporting plane for u. Theorem 1.5 shows that if the boundary data has quadratic growth near {x n = 0} then, each section of u at 0 is equivalent to a half-ellipsoid centered at 0. Theorem 1.5 (Localization theorem). Assume that Ω, u satisfy (1.6)-(1.8) above and, Then, for each h < c 0 there exists an ellipsoid E h of volume ω n 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 c 0 > 0 above depends on ρ, λ, Λ,and n.

Proof of Theorem 1.2
We start by remarking that under the assumptions of Theorem 1.5 above, we obtain that also the section {u < h 1/2 x n } has the shape of E h .
Indeed, since S h ⊂ c −1 0 E h ⊂ {x n ≤ c −1 0 h 1/2 } and u(0) = 0, we can conclude from the convexity of u that the set F := {x ∈ Ω| u < c 0 h 1/2 x n } satisfies for all small h and F is tangent to ∂Ω at 0. We show that F is equivalent to E h by bounding its volume by below.
Next we prove Theorem 1.2. We denote by c, C positive constants that depend on ρ, λ, Λ, p, n and ∂Ω C 1,1 , u| ∂Ω C 1,1 , u C 0,1 . For simplicity of notation, their values may change from line to line whenever there is no possibility of confusion.

Lemma 2.3 (Vitali covering). There exists a sequence of disjoint sections
where δ > 0 is a small constant that depends only on λ, Λ and n.
The existence of δ follows from the engulfing properties of sections of solutions to Monge-Ampere equation (1.2) with bounded right hand side (see [CG]). Now the proof is identical to the proof of Vitali's covering lemma for balls. We choose S δh1 (y 1 ) from all the sections S δh(y) (y), y ∈ Ω such that h(y 1 ) ≥ 1 2 sup yh (y), then choose S δh2 (y 2 ) as above but only from the remaining sections S δh(y) (y) that are disjoint from S δh1 (y 1 ), then S δh3 (y 3 ) etc. We easily obtain Sh i/2 (y i ).
End of proof of Theorem 1.2 There are a finite number of sections withh i ≥ c 1 and, by the interior W 2,p estimate, in each such section we have Sh i /2 (yi) D 2 u p dx ≤ C.
Next we consider the family F d of sections Sh i/2 (y i ) such that d/2 <h i ≤ d for some constant d ≤ c 1 . By Lemma 2.2 in each such section Sh i /2 (yi) D 2 u p dx ≤ C| log d| 2p |S δhi (y i )|, and since S δhi (y i ) ⊂ D Cd 1/2 are disjoint we find i∈F d Sh i /2 (yi) D 2 u p dx ≤ C| log d| 2p d 1/2 .
We add these inequalities for the sequence d = c 1 2 −k , k = 0, 1, . . . and obtain the desired bound.