Journal of the European Mathematical Society


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Volume 9, Issue 1, 2007, pp. 85–121free sample!
DOI: 10.4171/JEMS/74

Optimal mass transportation and Mather theory

Patrick Bernard[1] and Boris Buffoni[2]

(1) CEREMADE, Université de Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775, PARIS CEDEX 16, FRANCE
(2) Département Mathématiques, Ecole Polytechnique Federale, 1015, LAUSANNE, SWITZERLAND

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.

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Bernard Patrick, Buffoni Boris: Optimal mass transportation and Mather theory. J. Eur. Math. Soc. 9 (2007), 85-121. doi: 10.4171/JEMS/74