
Journal of Convex Analysis 15 (2008), No. 3, 593622 Copyright Heldermann Verlag 2008 Dynamic Formulation of Optimal Transport Problems Chloé Jimenez Lab. de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 Avenue Le Gorgeu, 29200 Brest, France chloe.jimenez@univbrest.fr [Abstractpdf] \def\R{\mathbb R} We consider the classical MongeKantorovich transport problem with a general cost $c(x,y)=F(yx)$ 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) 91121] 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) 139168] using an adaptation of Y. Brenier's dynamic formulation. Keywords: Wasserstein distance, optimal transport map, measure functionals, duality, tangential gradient, partial differential equations. MSC: 39B62, 46N10, 49Q20 [ Fulltextpdf (256 KB)] for subscribers only. 