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On the \(C^1\) and \(C^2\)-Convergence to Weak K.A.M. Solutions

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Abstract

We introduce a notion of upper Green regular solutions to the Lax-Oleinik semi-group that is defined on the set of \(C^0\) functions of a closed manifold via a Tonelli Lagrangian. Then we prove some weak \(C^2\) convergence results to such a solution for a large class of approximated solutions as

  • (1) the discounted solution (see [DFIZ16]);

  • (2) the image of a \(C^0\) function by the Lax-Oleinik semi-group;

  • (3) the weak K.A.M. solutions for perturbed cohomology class.

This kind of convergence implies the convergence in measure of the second derivatives. Moreover, we provide an example that is not upper Green regular and to which we have \(C^1\) convergence but not convergence in measure of the second derivatives.

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Notes

  1. We can have \(\mu _0=\infty \).

  2. See Definition 1 for the notion of \(C^1\)-convergence for functions that are not \(C^1\).

  3. \(\alpha (c)\) is Mañé critical value for the cohomology class c, see [Fat08].

  4. See Definition  1 for the notion of \(C^1\)-convergence for functions that are not \(C^1\).

  5. Here \(D^+u(x)\) denotes the set of super-differentials of u at x, see Section 2 for the definition.

  6. See Section 2 for the definition.

  7. See Section 2 for the notation.

  8. Here \(D^+u(x)\) denotes the set of super-differentials of u at x, see Section 2 for the definition.

  9. The existence of the limit is due to weak K.A.M. theorem, see [Fat08].

  10. Either this is \(+\infty \) or this belongs to \({\mathcal S}\) and is different from \(t_1\).

  11. Observe that it is proved in [Fat08] (Theorem 4.11.5) that every \(C^1\) weak K.A.M. solution is in fact \(C^{1, 1}\).

  12. This neighbourhood may be smaller than the neighbourhood that we introduced before.

  13. See the latest notation for the meaning of \(K^\lambda _q\).

  14. \(\mu \) can be a cohomology class, a discounted positive parameter or a positive time for the Lax–Oleinik semi-group.

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Acknowledgements

The authors thank the anonymous referees for their careful reading and for several suggestions that have improved the paper. X. Su is supported by the National Natural Science Foundation of China (Grant No. 11971060, 11871242) and M.-C. Arnaud is supported by the ANR (ANR AAPG 2021 PRC CoSyDy: ANR-CE40-0014).

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Authors and Affiliations

  1. CNRS, IMJ-PRG, Université de Paris Cité and Sorbonne Université, 75006, Paris, France

    Marie-Claude Arnaud

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (Ministry of Education), Beijing Normal University, No. 19, XinJieKouWai St., HaiDian District, Beijing, 100875, People’s Republic of China

    Xifeng Su

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  1. Marie-Claude Arnaud
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Correspondence to Xifeng Su.

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Communicated by C. Liverani.

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M.-C. Arnaud: member of the Institut universitaire de France.

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Arnaud, MC., Su, X. On the \(C^1\) and \(C^2\)-Convergence to Weak K.A.M. Solutions. Commun. Math. Phys. 392, 825–861 (2022). https://doi.org/10.1007/s00220-022-04355-4

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