Free boundary regularity in the optimal partial transport problem

TL;DR

This paper studies the regularity of free boundaries in the optimal partial transport problem, showing they are smooth hypersurfaces under certain conditions and estimating the size of singular sets.

Contribution

It improves previous regularity results, extends them to more general domains, and develops a method to estimate the Hausdorff dimension of singular sets.

Findings

Free boundaries are $C_{loc}^{1,eta}$ hypersurfaces away from a singular set.

The singular set has Hausdorff dimension at most $n-2$.

The singular set is finite in measure when domains are separated by a hyperplane.

Abstract

In the optimal partial transport problem, one is asked to transport a fraction $0<m \leq \min\{||f||_{L^1}, ||g||_{L^1}\}$ of the mass of $f=f \chi_\Omega$ onto $g=g\chi_\Lambda$ while minimizing a transportation cost. If $f$ and $g$ are bounded away from zero and infinity on strictly convex domains $\Omega$ and $\Lambda$, respectively, and if the cost is quadratic, then away from $\partial(\Omega \cap \Lambda)$ the free boundaries of the active regions are shown to be $C_{loc}^{1,\alpha}$ hypersurfaces up to a possible singular set. This improves and generalizes a result of Caffarelli and McCann \cite{CM} and solves a problem discussed by Figalli \cite[Remark 4.15]{Fi}. Moreover, a method is developed to estimate the Hausdorff dimension of the singular set: assuming $\Omega$ and $\Lambda$ to be uniformly convex domains with $C^{1,1}$ boundaries, we prove that the singular set is $\mathcal{H}^{n-2}$ $\sigma$-finite in the general case and $\mathcal{H}^{n-2}$ finite if $\Omega$ and $\Lambda$ are separated by a hyperplane.