# Boundary $\varepsilon$-regularity in optimal transportation

created by figalli on 12 Dec 2014
modified on 01 Jun 2017

[BibTeX]

Accepted Paper

Inserted: 12 dec 2014
Last Updated: 1 jun 2017

We develop an $\epsilon$-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 $-x\cdot y$.