Dynamics of Optimal Partial Transport

TL;DR

This paper investigates the evolution of free boundaries in optimal partial transport, providing estimates on their speed and showing how the transported mass relates to the cost cap, with a focus on quadratic costs.

Contribution

It offers new regularity estimates for free boundary motion and establishes Lipschitz continuity of the transported mass as a function of the cost cap.

Findings

Hölder and Lipschitz estimates on free boundary speed

Lipschitz continuity of mass with respect to cost cap

Enhanced understanding of parameter relationships in optimal partial transport

Abstract

This paper considers the evolution dynamics of the free boundaries in terms of the change of $m$, the allowed amount of transported mass or the change of $\lambda$, the transportation cost cap, i.e. the allowed maximum cost for a unit mass to be transported. Focusing on the quadratic cost function, we show H\"older and Lipschitz estimates on the speed of the free boundary motion in terms of $m$ and $\lambda$, respectively. It is also shown that the parameter $m$ is a Lipschitz function of $\lambda$, which previously was known only to be a continuous increasing function \cite{Ca-Mc}.