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Regularization of Subsolutions in Discrete Weak KAM Theory

Published online by Cambridge University Press:  20 November 2018

Patrick Bernard  and
Maxime Zavidovique
Patrick Bernard
Affiliation:
Département de mathématiques et applications, UMR CNRS 8553, Ecole Normale Supérieure, 45 rue d'Ulm, 75005, Paris, France, e-mail: [email protected]
Maxime Zavidovique
Affiliation:
Institut de mathématiques de Jussieu, UMR CNRS 8553, Université Pierre et Marie Curie, Case 247, 4, Place Jussieu, UMR CNRS 7586, Paris, France, e-mail: [email protected]
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Abstract

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We expose different methods of regularizations of subsolutions in the context of discrete weak $\text{KAM}$ theory that allow us to prove the existence and the density of ${{C}^{1,1}}$ subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.

Keywords

49C15discrete subsolutionsregularity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 740 - 756
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Ambrosio, L. and Dancer, N., Calculus of variations and partial differential equations. Topics on geometrical evolution problems and degree theory. Papers from the Summer School held in Pisa, September 1996. Springer-Verlag, Berlin, 2000.Google Scholar
[2] Bernard, P., Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier (Grenoble) 52(2002), no. 5, 1533–1568. http://dx.doi.org/10.5802/aif.1924 Google Scholar
[3] Bernard, P., Existence of C1,1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds. Ann. Sci. École Norm. Sup. (4) 40(2007), no. 3, 445–452.Google Scholar
[4] Bernard, P., The dynamics of pseudographs in convex Hamiltonian systems. J. Amer. Math. Soc. 21(2008), no. 3, 615–669. http://dx.doi.org/10.1090/S0894-0347-08-00591-2 http://dx.doi.org/10.1090/S0894-0347-08-00591-2 Google Scholar
[5] Bernard, P., Lasry-Lions regularization and a lemma of Ilmanen. Rend. Semin. Mat. Univ. Padova 124(2010), 221–229.Google Scholar
[6] Bernard, P. and Buffoni, B., Weak KAM pairs and Monge-Kantorovich duality. In: Asymptotic analysis and singularities—elliptic and parabolic PDEs and related problems, Adv. Stud. Pure Math., 4-27, Math. Soc. Japan, Tokyo, 2007, pp. 397–420.Google Scholar
[7] Bernard, P. and Roquejoffre, J. M., Convergence to time-periodic solutions in time-periodic Hamilton–Jacobi equations on the circle. Comm. Partial Differential Equations 29(2004), no. 3–4, 457–469. http://dx.doi.org/10.1081/PDE-120030404 Google Scholar
[8] Cardaliaguet, P., Front propagation problems with nonlocal terms. II. J. Math. Anal. Appl. 260(2001), no. 5, 572–601. http://dx.doi.org/10.1006/jmaa.2001.7483 Google Scholar
[9] Constantine, G. M. and Savits, T. H., A multivariate Faà di Bruno formula with applications. Trans. Amer. Math. Soc. 348(1996), no. 2, 503–520. http://dx.doi.org/10.1090/S0002-9947-96-01501-2 http://dx.doi.org/10.1090/S0002-9947-96-01501-2 Google Scholar
[10] Contreras, G. and Iturriaga, R., and Sanchez-Morgado, H., Weak solutions of the Hamilton-Jacobi equation for time periodic Lagrangians. preprint. http://www.cimat.mx/_gonzalo/papers/whj.pdf Google Scholar
[11] de Rham, G., Variétés différentiables. Formes, courants, formes harmoniques. Troisiàme édition revue et augmentée, Publications de l’Institut de Mathématique de l’Université de Nancago, III, Actualités Scientifiques et Industrielles, 1222b, Hermann, Paris, 1973.Google Scholar
[12] Fathi, A., Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327(1998), no. 3, 267–270. http://dx.doi.org/10.1016/S0764-4442(98)80144-4 Google Scholar
[13] Fathi, A. and Mather, J., Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case. Bull. Soc. Math. France 128(2000), no. 3, 473–483.Google Scholar
[14] Fathi, A. and Siconolfi, A., Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155(2004), no. 2, 363–388. http://dx.doi.org/10.1007/s00222-003-0323-6 Google Scholar
[15] Fathi, A. and Zavidovique, M., Ilmanen's lemma on insertion of C1,1 functions. Rend. Semin. Mat. Univ. Padova 124(2010), 203–219.Google Scholar
[16] Gomes, D. A., Viscosity solution method and the discrete Aubry-Mather problem. Discrete Contin. Dyn. Syst. 13(2005), no. 1, 103–116. http://dx.doi.org/10.3934/dcds.2005.13.103 Google Scholar
[17] Hirsch, M.W., Differential topology. Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994.Google Scholar
[18] Ilmanen, T., The level-set flow on a manifold. In: Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, American Mathematical Society, Providence, RI, 1993, pp. 193–204.Google Scholar
[19] Zavidovique, M., Strict sub-solutions and Ma˜ñé potential in discrete weak KAM theory. Comment. Math. Helv. 87(2012), no. 1, 1–39. http://dx.doi.org/10.4171/CMH/247 Google Scholar
[20] Zavidovique, M., Existence of C1,1 critical subsolutions in discrete weak KAM theory. J. Mod. Dyn. 4(2010), no. 4, 693–714. http://dx.doi.org/10.3934/jmd.2010.4.693 http://dx.doi.org/10.3934/jmd.2010.4.693 Google Scholar