Dynamics of Optimal Partial Transport

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}.