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Minimal measures, one-dimensional currents and the Monge–Kantorovich problem

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Abstract

In recent works L.C. Evans has noticed a strong analogy between Mather's theory of minimal measures in Lagrangian dynamic and the theory developed in the last years for the optimalmass 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. Someminimality properties of such curves are also addressed.

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Pisa, via Bonanno Pisano 25/B, 56126, Pisa, Italy

    Luigi De Pascale

  2. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Maria Stella Gelli

  3. Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125, Bari, Italy

    Luca Granieri

Authors
  1. Luigi De Pascale
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  2. Maria Stella Gelli
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  3. Luca Granieri
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Correspondence to Luigi De Pascale.

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Mathematics Subject Classification (2000)37J50, 49Q20, 49Q15

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De Pascale, L., Gelli, M.S. & Granieri, L. Minimal measures, one-dimensional currents and the Monge–Kantorovich problem. Calc. Var. 27, 1–23 (2006). https://doi.org/10.1007/s00526-006-0017-1

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  • DOI: https://doi.org/10.1007/s00526-006-0017-1

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