Global regularity for the Monge-Ampère equation with natural boundary condition

Abstract

In this paper, we establish the global $C^{2,\alpha}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\mathrm{det}\ D^2u = f$ subject to boundary condition $Du(\Omega) = \Omega^\ast$, where $\Omega$ and $\Omega^\ast$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.