Global $C^{2,\alpha}$ estimates for the Monge-Amp\`ere equation on polygonal domains in the plane

TL;DR

This paper classifies solutions of the Monge-Ampère equation on the first quadrant with quadratic boundary conditions and derives global regularity estimates for convex polygonal domains assuming a suitable subsolution exists.

Contribution

It provides a classification of solutions on the first quadrant and establishes global $C^{2, heta}$ estimates for the Monge-Ampère equation on convex polygons in the plane.

Findings

Classification of solutions with quadratic boundary data

Global $C^{2, heta}$ estimates for convex polygons

Existence of subsolutions ensures regularity

Abstract

We classify global solutions of the Monge-Amp\`ere equation $\det D^2 u=1 $ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^{2,\alpha}$ estimates for the non-degenerate Monge-Amp\`ere equation in convex polygonal domains in $\mathbb R^2$ provided a globally $C^2$, convex strict subsolution exists.