Regularity of singular set in optimal transportation

TL;DR

This paper develops a new regularity theory for the singular set in optimal transportation between a source and a target with two disjoint convex domains, achieving higher order regularity without source convexity.

Contribution

It introduces novel methods to establish higher order regularity of the singular set in optimal transport problems with disjoint convex targets, independent of source convexity.

Findings

Higher order regularity of the singular set is proven.

Methods developed have broader applications beyond the specific problem.

Results align with and extend Caffarelli's regularity theory.

Abstract

In this paper, we establish a regularity theory for the optimal transport problem when the target is composed of two disjoint convex domains. This is an important model in which singularities arise. Even though the singular set does not exhibit any form of convexity a priori, we prove its higher order regularity by developing novel methods, which also have many other applications. Notably, our results are achieved without requiring any convexity of the source domain. This aligns with Caffarelli's celebrated regularity theory.