On action minimizing measures for the Monge-Kantorovich problem

Abstract

In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological constraints of Mather’s problem by boundary terms and we investigate the equivalence with the mass transportation problem. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier’s formulation are also established.