Calculus of Variations and Geometric Measure Theory

L. De Pascale - M. S. Gelli - L. Granieri

Minimal measures, one-dimensional currents and the Monge-Kantorovich problem

created by depascal on 25 Nov 2004
modified by granieri on 06 Sep 2010


Published Paper

Inserted: 25 nov 2004
Last Updated: 6 sep 2010

Journal: Calc. Var. Partial Differential Equations
Volume: 27
Number: 1
Pages: 1-23
Year: 2006

The original paper is available on http:/www.springerlink.comcontentd0l14v304n6n3457


In recent works L.C. Evans has noticed a strong analogy between Mather's theory of minimal measures in Lagrangian dynamic and theory developed in the last years for the optimal mass transportation (or Monge-Kantorovich) problem. In this paper we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Some minimality properties of such curves are also addressed.

Keywords: Mather's minimal measures, Monge-Kantorovich problem, normal 1-currents, optimal transport problems