Partial regularity of optimal transport with Coulomb cost

TL;DR

This paper proves that for two-marginal optimal transport problems with Coulomb cost, the optimal map is smooth outside a measure-zero set under certain regularity conditions on the marginals, despite known singularities.

Contribution

It establishes partial regularity of optimal transport maps with Coulomb cost, showing smoothness outside a negligible set under mild regularity assumptions.

Findings

Optimal maps are $C^{1,eta}$ outside a measure-zero set.

Singularities occur across codimension 1 surfaces, even with smooth marginals.

Regularity holds when marginals are Hölder continuous and bounded away from zero and infinity.

Abstract

We prove that for two-marginal optimal transport with Coulomb cost, 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 and bounded away from zero and infinity. 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).