Regularity of Optimal Transport for Rough Measures

Abstract

In this thesis we study the regularity properties of solutions to the Kantorovich optimal transportation problem for the quadratic cost between two measures that are not necessarily absolutely continuous. More concretely, we only assume that the two measures have a property analogous to absolute continuity, but only at a certain macroscopic scale that can be different for each of them. Using a local description of optimal transportation, we prove that Kantorovich potentials cannot fail to be strictly convex at scales larger than the macroscopic scales at which the measures are absolutely continuous. Using an argument based on duality we prove a $C^1$ regularity result for the conjugate of the potential up to the scales of the measures. Our proof does not use the classical regularity theory for the Monge-Amp{`e}re equation. Instead, it relies on direct estimates for the bounds of an integral quantity involving the Kantorovich potential, that are based on optimal transportation arguments.