Optimal transport maps in Monge-Kantorovich problem

Details Optimal transport maps in Monge-Kantorovich problem Optimal transport maps in Monge-Kantorovich problem

2003-04-24

arxiv.org/abs/math/0304389v1

Proceedings of the ICM, Beijing 2002, vol. 3, 131--140

Mathematics

Analysis of PDEs

Scientific paper

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.

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