Optimal transport maps in Monge-Kantorovich problem

TL;DR

This paper reviews the classical Monge-Kantorovich optimal transport problem and discusses recent advances in establishing the existence of optimal transport maps using singular perturbation and $ extGamma$-convergence techniques.

Contribution

It introduces new methods for proving the existence of optimal transport maps, extending classical results through singular perturbation and $ extGamma$-convergence approaches.

Findings

Existence of optimal transport maps in several cases.

Stability results for classical Monge solutions.

Application of $ extGamma$-convergence to optimal transport.

Abstract

In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence of optimal transport maps. We show that in several cases optimal transport maps can be obtained by a singular perturbation technique based on the theory of $\Gamma$-convergence, which yields as a byproduct existence and stability results for classical Monge solutions.