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 the Monge-Ampère equation ${\mathrm{det}}\, D^2u = f$ 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 Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.


Shibing Chen

School of Mathematical Sciences, University of Science and Technology of China, Hefei, P.R. China

Jiakun Liu

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia

Xu-Jia Wang

Centre for Mathematics and Its Application, The Australian National University, Canberra, Australia