Abstract

We consider the classical Monge-Kantorovich transport problem with a general cost c(x,y)=F(y−x)c(x,y)=F(y-x) where F ⁣:Rd→R+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. Such a characterization has been given in the smooth case by L. Evans and W. Gangbo [Mem. Amer. Math. Soc. 653 (1999)] where FF is the Euclidian norm and by Y. Brenier [Lecture Notes Math. 1813 (2003) 91–121] in the case where F=∣⋅∣pF=\vert \cdot \vert^p with p>1p>1. We extend these results to the case of general FF 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Chloé Jimenez

Lab. de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 Avenue Le Gorgeu, 29200 Brest, France

chloe.jimenez@univ-brest.fr

Keywords

  • Wasserstein distance
  • optimal transport map
  • measure functionals
  • duality
  • tangential gradient
  • partial differential equations

Mathematics Subject Classification

39B62, 46N10, 49Q20

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Suggested citation

C. Jimenez. “Dynamic Formulation of Optimal Transport Problems.” Journal of Convex Analysis 15 (2008), No. 3, 593–622. https://doi.org/10.68381/jca15042

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.