Global $C^{1,\alpha}$ regularity for Monge-Amp\`ere equations on planar convex domains

TL;DR

This paper proves that solutions to the Monge-Ampère equation on convex domains have globally Hölder continuous gradients, extending regularity results beyond uniformly convex domains.

Contribution

It establishes the first global $C^{1,eta}$ regularity for solutions on strictly convex, non-uniformly convex domains.

Findings

Proves global Hölder gradient estimates for Monge-Ampère solutions

Extends regularity results to non-uniformly convex domains

Provides new techniques for boundary regularity analysis

Abstract

In this paper, we establish the global H\"older gradient estimate for solutions to the Dirichlet problem of the Monge-Amp\`ere equation $\det D^2u = f$ on strictly convex but not uniformly convex domain $\Omega$.