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

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.