Continuity and estimates for multimarginal optimal transportation problems with singular costs

TL;DR

This paper investigates regularity and continuity properties of solutions in multimarginal optimal transportation problems with singular costs, including Coulomb interactions, providing new estimates and regularity results.

Contribution

It establishes regularity of minimizers and dual maximizers, and derives estimates and continuity properties for problems with singular costs like Coulomb.

Findings

Regularity of optimal transportation plans

Existence and regularity of Kantorovich potentials

Cost estimates and continuity properties

Abstract

We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.