Global regularity for the Monge-Amp\`ere equation with natural boundary condition

Abstract

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