Partial regularity of optimal transport with Coulomb cost

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 $\alpha$-Hölder 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).

20 pages