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

Journal Article

### 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.

• 2018

### 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.

• 2018