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Two remarks about Mañé’s conjecture. (English) Zbl 1209.37068

Summary: We consider R. Mañé’s conjectures and prove that the one he made in [Bol. Soc. Bras. Mat., Nova Sér. 28, No. 2, 141–153 (1997; Zbl 0892.58064)] is stronger than the one he made in [Nonlinearity 9, No. 2, 273–310 (1996; Zbl 0886.58037)]. Then, we prove that the most straightforward approach to prove the strong conjecture does not work in the \(C^{4}\) topology.

MSC:

37J50 Action-minimizing orbits and measures (MSC2010)
70H20 Hamilton-Jacobi equations in mechanics
35F20 Nonlinear first-order PDEs
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

[1] Mañé, R., Lagrangian Flows: the Dynamics of Globally Minimizing Orbits, Bol. Soc. Brasil. Mat. (N.S.), 1997, vol. 28, no. 2, pp. 141–153. · Zbl 0892.58064 · doi:10.1007/BF01233389
[2] Mañé, R., Generic Properties and Problems of Minimizing Measures of Lagrangian Systems, Nonlinearity, 1996, vol. 9, no. 2, pp. 273–310. · Zbl 0886.58037 · doi:10.1088/0951-7715/9/2/002
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[8] Figalli, A. and Rifford, L., Closing Aubry Sets, preprint; http://math.unice.fr/rifford . · Zbl 1377.37090
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