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

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