Regularity of free boundaries in optimal transportation

TL;DR

This paper studies the smoothness of free boundaries in optimal transportation problems with quadratic cost, establishing conditions under which these boundaries are differentiable and possess higher regularity.

Contribution

It proves $C^{1,eta}$ and $C^{2,eta}$ regularity results for free boundaries in optimal transport, extending boundary regularity of Monge-Ampère equations.

Findings

Free boundaries are $C^{1,eta}$ for convex domains with bounded densities.

Free boundaries are $C^{2,eta}$ when densities are $C^eta$ and domains are sufficiently smooth.

Regularity results apply to problems with two separate target sets.

Abstract

In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,\alpha}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^\alpha$, and $\partial\Omega, \partial\Omega^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Amp\`ere equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,\alpha}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets.