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arXiv · 1606.05173

Partial $W^{2,p}$ regularity for optimal transport maps

Shibing Chen·Alessio Figalli
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Abstract

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.

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Shibing Chen, Alessio Figalli. 2016-06-16. Partial $W^{2,p}$ regularity for optimal transport maps. https://arxiv.org/abs/1606.05173

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