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

Shibing Chen; Jiakun Liu; Xu-Jia Wang

doi:https://doi.org/10.4007/annals.2021.194.3.4

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.

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

Abstract

Authors