Sharp boundary ε-regularity of optimal transport maps

Boundary Ε-regularity in Optimal Transportation

We develop an ε-regularity theory at the boundary for a general class of MongeAmpère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on C uniformly convex domains are C up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x · y.

Regularity of Optimal Transport Maps

In the special case “cost=squared distance” on R, the problem was solved by Caffarelli [Caf1, Caf2, Caf3, Caf4], who proved the smoothness of the map under suitable assumptions on the regularity of the densities and on the geometry of their support. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case “cost=squared distance”...

Regularity of optimal transport maps and applications

Boundary Regularity of Optimal Transport Paths

The optimal transport problem aims at finding an optimal way to transport a given probability measure into another. In contrast to the wellknown Monge-Kantorovich problem, the ramified optimal transportation problem aims at modeling a tree-typed branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to fa...

Partial Regularity for Optimal Transport Maps

We prove that, for general cost functions on R, or for the cost d/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.