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 $λ$, 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 Hölder and Lipschitz estimates on the speed of the free boundary motion in terms of $m$ and $λ$, respectively. It is also shown that the parameter $m$ is a Lipschitz function of $λ$, which previously was known only to be a continuous increasing function \cite{Ca-Mc}.

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