Local regularity result for an optimal transportation problem with rough measures in the plane

TL;DR

This paper establishes local regularity properties of convex functions related to optimal transportation between measures that are not fully smooth, using novel estimates based on transportation problems rather than classical PDE methods.

Contribution

It introduces a discrete scale concept for measures and proves regularity results for convex functions and their Legendre transforms in this setting.

Findings

Convex functions cannot have flat parts larger than the discrete scale.

Legendre transforms are $C^1$ up to the discrete scale.

Results apply to transportation between measures composed of Dirac masses.

Abstract

We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result up to the discrete scale for the Legendre transform. Our result applies in particular to any Kantorovich potential associated to an optimal transportation problem between two measures that are (possibly only locally) sums of uniformly distributed Dirac masses. The proof relies on novel explicit estimates directly based on the optimal transportation problem, instead of the Monge-Ampere equation.