ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2011 - Volume 17, Issue 03  

Research Article

Local semiconvexity of Kantorovich potentials on non-compact manifolds*

Figalli, Alessioa1 and Gigli, Nicolaa2

a1 Centre de Mathématiques Laurent Schwartz, UMR 7640, École Polytechnique, 91128 Palaiseau, France. figalli@math.polytechnique.fr

a2 University of Bordeaux, France. nicolagigli@googlemail.com

Abstract

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.

(Received July 4 2009)

(Revised October 13 2009)

(Online publication March 31 2010)

Key Words:

  • Kantorovich potential;
  • optimal transport;
  • regularity

Mathematics Subject Classification:

  • 49Q20;
  • 35J96

Footnotes

*  N. Gigli was partially financed by KAM Faible, ANR-07-BLAN-0361.