Weak KAM pairs and Monge-Kantorovich duality

Details Weak KAM pairs and Monge-Kantorovich duality Weak KAM pairs and Monge-Kantorovich duality

2005-11-30

arxiv.org/abs/math/0511746v2

Mathematics

Dynamical Systems

Advanced Studies in Pure Mathematics 47, 2 (2007)

Scientific paper

The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way.

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