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arXiv · 2508.01756

Partial regularity of optimal transport with Coulomb cost

Gero Friesecke·Tobias Ried
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Abstract

We prove that for two-marginal optimal transport with Coulomb cost on $\mathbb{R}^d$, the optimal map is a $C^{1,\alpha}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $\alpha$-H\"older continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains).

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Gero Friesecke, Tobias Ried. 2025-08-03. Partial regularity of optimal transport with Coulomb cost. https://arxiv.org/abs/2508.01756

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