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arXiv · 1303.2715

On the regularity of the free boundary in the optimal partial transport problem

Shibing Chen·Emanuel Indrei
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Abstract

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some $L^p(\mathbb{R}^n)$ space for $p \in (\frac{n+1}{2},\infty]$ and the positive target density is bounded away from zero, then the free boundary is a semiconvex $C_{loc}^{1,\alpha}$ hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.

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Shibing Chen, Emanuel Indrei. 2013-03-11. On the regularity of the free boundary in the optimal partial transport problem. https://arxiv.org/abs/1303.2715

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