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On global $W^{2,δ}$ estimates for the Monge-Ampère equation on general bounded convex domains

Nam Q. Le

Abstract

We establish global $W^{2,δ}$ estimates, for all $δ<\frac{1}{n-1}$, for convex solutions to the Monge-Ampère equation with positive $C^{2,β}$ right-hand side and zero boundary values on general bounded convex domains in ${\mathbb R}^n$ ($n\geq 2$). We exhibit examples showing that global $W^{2, \frac{n}{2(n-1)}}$ estimates fail in all dimensions, so the range of $δ$ is sharp in two dimensions.

On global $W^{2,δ}$ estimates for the Monge-Ampère equation on general bounded convex domains

Abstract

We establish global estimates, for all , for convex solutions to the Monge-Ampère equation with positive right-hand side and zero boundary values on general bounded convex domains in (). We exhibit examples showing that global estimates fail in all dimensions, so the range of is sharp in two dimensions.
Paper Structure (5 sections, 6 theorems, 64 equations)

This paper contains 5 sections, 6 theorems, 64 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Proof of Theorem \ref{['W2del_thm']}
  3. Global $W^{2,\delta}$ estimates
  4. The rectangular box domain
  5. Further remarks

Key Result

Theorem 1.1

Let $u\in C(\overline{\Omega})$ be the convex Aleksandrov solution to the Monge-Ampère equation MA1 where $\Omega$ is a bounded convex domain in $\mathbb R^n$ ($n\geq 2$), and $f\in C^{2,\beta}(\overline\Omega)$ satisfies fbd where $\beta\in (0, 1)$. Then the following statements hold.

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['D2_ptw']}
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 4 more