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Optimal mass transportation and Mather theory. (English) Zbl 1241.49025
Summary: We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.

49Q20 Variational problems in a geometric measure-theoretic setting
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