Dynamic Formulation of Optimal Transport Problems

We consider the classical Monge-Kantorovich transport problem with a general cost

where

F \colon {\mathbb R}^d \to {\mathbb R}^+ F  ⁣ : R d → 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. Such a characterization has been given in the smooth case by L. Evans and W. Gangbo [Mem. Amer. Math. Soc. 653 (1999)] where

F 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 F = ∣ ⋅ ∣ p

with

p>1 p > 1

. We extend these results to the case of general

F F

and singular transported measures in the spirit of previous work by G. Bouchitté and G. Buttazzo [J. Eur. Math. Soc. 3 (2001) 139–168] using an adaptation of Y. Brenier's dynamic formulation.