A Strengthened Alexandrov Maximum Principle or Uniform H\"older Continuity for Solutions of the Monge--Amp\`ere Equation with Bounded Right-Hand Side

TL;DR

This paper establishes optimal H"older continuity bounds for convex solutions of the Monge--Ampère equation with bounded right-hand side, extending the Alexandrov maximum principle to higher dimensions.

Contribution

It proves sharp H"older continuity estimates for solutions, with explicit exponents depending on the dimension, using Hessian determinant bounds and comparison principles.

Findings

H"older exponent =2/n for n

H"older exponent  for n=2

Bounds are proven to be sharp

Abstract

This article is about the convex solution $u$ of the Monge--Amp\`ere equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)| \leq C \operatorname{dist}(x,\partial\Omega)^\alpha$ is equivalent to (uniform) H\"older continuity with the same constant and exponent. Convex $\alpha$-H\"older continuous functions are $W^{1,p}$ for $p < 1/(1{-}\alpha)$. We prove H\"older continuity with the exponent $\alpha=2/n$ for $n \geq 3$ and any $\alpha \in (0,1)$ for $n=2$, provided that the boundary data satisfy this H\"older continuity, and show that these bounds for the exponent are sharp. The only means is to bound the Hessian determinant of a certain explicit function on an $n$-dimensional cylinder and to use the comparison princple.