Regularity of free boundaries in optimal transportation

Shibing Chen; Jiakun Liu

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arXiv · 1911.10790

Regularity of free boundaries in optimal transportation

Shibing Chen·Jiakun Liu

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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.

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Shibing Chen, Jiakun Liu. 2019-11-25. Regularity of free boundaries in optimal transportation. https://arxiv.org/abs/1911.10790

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