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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.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
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References:
[1] Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces (Funchal, 200), Lecture Notes in Math. 1812, Springer, 1-52 (2003). · Zbl 1047.35001
[2] Ambrosio, L.: Lecture notes on transport equation and Cauchy problem for BV vector fields and applications. (2004) · Zbl 1075.35087
[3] Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Math. ETH Zürich, Birkhäuser (2005) · Zbl 1090.35002
[4] Ambrosio, L., Pratelli, A.: Existence and stability results in the L1 theory of optimal trans- portation. In: Lecture Notes in Math. 1813, Springer, 123-160 (2003) · Zbl 1065.49026
[5] Bangert, V.: Minimal measures and minimizing closed normal one-currents. Geom. Funct. Anal. 9 , 413-427 (1999) · Zbl 0973.58004
[6] Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge- Kantorovich mass transfer problem. Numer. Math. 84 , 375-393 (2000) · Zbl 0968.76069
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