An overview on flows of vector fields and on optimal transport

Abstract

The aim of this Chapter is twofold. On one side, we give an overview on the classical results regarding flows of vector fields, the regularity of degenerate elliptic PDEs and, in particular, the Monge-Ampere equation. These results and ideas will be fundamental for the development of all the subsequent chapters. On the other side, we present the classical theory according to a point of view that will be useful in the rest of this thesis, showing refinements of the known theorems that suit the subsequent discussions.