Dynamic Formulation of Optimal Transport Problems

\def\R{\mathbb R} We consider the classical Monge-Kantorovich transport problem with a general cost $c(x,y)=F(y-x)$ where $F \colon \R^d \to \R^+$ is a convex function and our aim is to characterize the dual optimal potential as the solution of a system of partial differential equations. \par Such a characterization has been given in the smooth case by L. Evans and W. Gangbo [Mem. Amer. Math. Soc. 653 (1999)] where $F$ is the Euclidian norm and by Y. Brenier [Lecture Notes Math. 1813 (2003) 91--121] in the case where $F=\vert \cdot \vert^p$ with $p>1$. We extend these results to the case of general $F$ and singular transported measures in the spirit of previous work by G. Bouchitt\'e and G. Buttazzo [J. Eur. Math. Soc. 3 (2001) 139--168] using an adaptation of Y. Brenier's dynamic formulation.