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

Abstract

We classify global solutions of the Monge-Ampère 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-Ampère equation in convex polygonal domains in ${\mathbb{R}}^2$ provided a globally $C^2$, convex strict subsolution exists.

v2: final version incorporating suggestions from the referee report; to be published in Amer. J. Math