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

Abstract

In this paper, we establish the global $C^{2,\alpha }$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\det D^{2}u=f$ subject to boundary condition $Du(\Omega )={\Omega }^{\ast }$, where $\Omega$ and ${\Omega }^{\ast }$ 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.

The exposition is polished and references are updated

References in corpus (3)

• Mode Collapse and Regularity of Optimal Transportation Maps
• Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two
• $C^{2,\alpha }$ regularity of free boundaries in optimal transportation

Cited by in corpus (6)

• Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two
• Sharp boundary $\epsilon$-regularity of optimal transport maps
• On the Regularity of Optimal Transports between Degenerate Densities
• $C^{2,\alpha }$ regularity of free boundaries in optimal transportation
• Regularity of free boundaries in optimal transportation
• Global $W^{2,1+\epsilon }$ estimates for Monge-Ampère equation with natural boundary condition