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On the convergence of the Lax-Oleinik semigroup. (Sur la convergence du semi-groupe de Lax-Oleinik.) (French) Zbl 1052.37514

Author’s summary: We show the convergence of the Lax-Oleinik semigroup for a Lagrangian defined on the tangent space of a compact manifold which is strictly convex and superlinear in the fibers.” The author’s result is based on a ‘weak KAM theorem’ [A. Fathi, C. R. Acad. Sci.; Paris; Sér. I 324, No. 9, 1043–1046 (1997; Zbl 0885.58022)] and generalizes a result by J.-M. Roquejoffre [C. R. Acad. Sci., Paris, Sér. I 326, No. 2, 185–189 (1998; Zbl 0924.70017)]. The applications include the “Peierls barriers” of Aubry-Mather theory.

MSC:

37J50 Action-minimizing orbits and measures (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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