Partial regularity of optimal transport with Coulomb cost
Abstract
Abstract We prove that for two-marginal optimal transport with Coulomb cost on $$\mathbb {R}^d$$ R d , the optimal map is a $$C^{1,\alpha }$$ C 1 , α diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $$\alpha $$ α -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).
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Geometric Analysis and Curvature Flows
Point processes and geometric inequalities
Publication details
- Journal / source
- Calculus of Variations and Partial Differential Equations
- Volume
- 65
- Issue
- 7
- Publication date
- 26 June 2026
- Language
- EN