Past lectures

Abstract

The lectures will cover classical and recent results on the Monge-Kantorovich theory of optimal transport. In particular we focus on the problem of the existence of optimal transport maps and on the different formulations of the problem, due to Kantorovich, Benamou-Brenier and Evans-Gangbo. In the case of distance cost functions we also discuss the role of the so-called transport density and we analyze its regularity properties. We discuss also some applications of the theory to Shape Optimization and Partial Differential Equations.

Tentative index

1. Examples

2. Transport plans: existence and regularity

3. The one-dimensional case

4. The ODE version of the optimal transport problem

5. The PDE version of the optimal transport problem and the p-Laplacian approximation

6. Existence of optimal transport maps

7. Regularity and uniqueness of the transport density

8. Applications: (a) the mass optimization problem; (b) implicit time discretization and the porous medium equation.