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 C2C2 uniformly convex domains are C1,αC1,α up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x⋅y−x⋅y.