Boundary $\varepsilon$-regularity in optimal transportation

TL;DR

This paper develops a boundary regularity theory for optimal transportation problems, showing that under certain conditions, transport maps are smoothly regular up to the boundary.

Contribution

It introduces an $ ext{e}$-regularity framework at the boundary for Monge-Ampère equations in optimal transport, extending regularity results to boundary points.

Findings

Optimal transport maps are $C^{1,eta}$ up to the boundary under specified conditions.

Boundary regularity holds for transport maps between H"older densities on convex domains.

The theory applies when the cost function is a small perturbation of the quadratic cost.

Abstract

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$ uniformly convex domains are $C^{1,\alpha}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.