Calculus of Variations and Geometric Measure Theory
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L. Granieri

On Action Minimizing Measures for the Monge-Kantorovich Problem

created by granieri on 01 Dec 2004
modified on 06 Sep 2010


Published Paper

Inserted: 1 dec 2004
Last Updated: 6 sep 2010

Journal: NoDea
Volume: 14
Pages: 125-152
Year: 2007

The original paper is available on http:/www.springer.combirkhausermathematicsjournal30


\begin{abstract}In recent years different authors 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 clousure and homological constraints of Mather's problem with boundary terms and we invesigate the equivalence with the mass transportation problem. %alence in general fails for superlinear Lagrangians. Actually, we % prove that the mass transportation problem is a constrained version of such problem % which recover the original Mather's problem in the case of zero boundary term. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier's formulation are also established.


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


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