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Minimal measures, one-dimensional currents and the Monge-Kantorovich problem. (English) Zbl 1096.37033

Summary: In recent works, L.C. Evans has noticed a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamics and the theory developed in the last years for the optimal mass transportation (or Monge-Kantorovich) problem. In this paper, we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Some minimality properties of such curves are addressed, too.

MSC:

37J50 Action-minimizing orbits and measures (MSC2010)
49Q20 Variational problems in a geometric measure-theoretic setting
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
32U40 Currents
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