Global regularity for the Monge-Ampère equation with natural boundary condition
Pages 745-793 from Volume 194 (2021), Issue 3 by Shibing Chen, Jiakun Liu, Xu-Jia Wang
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 $Du(\Omega ) = \Omega ^*$, 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.
Received: 19 September 2019
Revised: 9 May 2021
Accepted: 13 August 2021
Published online: 2November 2021
Authors
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