Boundary H\"older gradient estimates for the Monge-Amp\`ere equation

TL;DR

This paper establishes global H"older gradient estimates for solutions to the Monge-Ampère equation under conditions on the domain's convexity and bounds on the right-hand side function.

Contribution

It provides new boundary gradient regularity results for the Monge-Ampère equation in both uniformly convex and flat domains.

Findings

Gradient estimates depend on domain convexity

Results hold for bounded away from zero and infinity right-hand side

Enhances understanding of boundary regularity for Monge-Ampère solutions

Abstract

We investigate global H\"older gradient estimates for solutions to the Monge-Amp\`ere equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main situations when a) the domain $\Omega$ is uniformly convex and b) $\Omega$ is flat.