Boundary ε-regularity in optimal transportation

We develop an ε-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 C2 uniformly convex domains are C1,α up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x · y.The second author has been partially supported by NSF Grant DMS-1262411. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000