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arXiv · 2002.08668

Sharp boundary $\varepsilon$-regularity of optimal transport maps

Tatsuya Miura·Felix Otto
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Abstract

In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,\alpha}$-boundary. Our main result asserts sharp $C^{1,\alpha}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,\alpha}$. Our method is completely variational and builds on the recently developed interior regularity theory.

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Tatsuya Miura, Felix Otto. 2020-02-20. Sharp boundary $\varepsilon$-regularity of optimal transport maps. https://doi.org/10.1016/j.aim.2021.107603

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