On the regularity of the free boundary in the optimal partial transport problem

TL;DR

This paper investigates the regularity and geometric properties of the free boundary in the optimal partial transport problem, establishing conditions under which the boundary is Lipschitz, semiconvex, or rectifiable, with applications to Riemannian geometry.

Contribution

It proves that a $C^1$ cost function yields a locally Lipschitz free boundary and extends regularity results to Riemannian settings, addressing open problems in the field.

Findings

A $C^1$ cost function implies a locally Lipschitz free boundary.

Under certain density and cost conditions, the free boundary is a semiconvex $C_{loc}^{1,eta}$ hypersurface.

A locally Lipschitz cost function leads to a rectifiable free boundary.

Abstract

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some $L^p(\mathbb{R}^n)$ space for $p \in (\frac{n+1}{2},\infty]$ and the positive target density is bounded away from zero, then the free boundary is a semiconvex $C_{loc}^{1,\alpha}$ hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.