Boundary $\epsilon$-regularity in optimal transportation

Abstract

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder 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 $-x\cdot y$.

24 pages, accepted by Advances in Mathematics