Geodesic
Regularization of Subsolutions in Discrete Weak KAM Theory
Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 740-756

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2012-059-3
Mots-clés : 49C15, discrete subsolutions, regularity 
Bernard, Patrick; Zavidovique, Maxime. Regularization of Subsolutions in Discrete Weak KAM Theory. Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 740-756. doi: 10.4153/CJM-2012-059-3
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[1] [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] [2] Bernard, P., Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier (Grenoble) 52(2002), no. 5, 1533–1568. Google Scholar | DOI

[3] [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] [4] Bernard, P., The dynamics of pseudographs in convex Hamiltonian systems. J. Amer. Math. Soc. 21(2008), no. 3, 615–669. Google Scholar | DOI

[5] [5] Bernard, P., Lasry-Lions regularization and a lemma of Ilmanen. Rend. Semin. Mat. Univ. Padova 124(2010), 221–229. Google Scholar

[6] [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] [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. Google Scholar | DOI

[8] [8] Cardaliaguet, P., Front propagation problems with nonlocal terms. II. J. Math. Anal. Appl. 260(2001), no. 5, 572–601. Google Scholar | DOI

[9] [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. Google Scholar | DOI

[10] [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] [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] [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. Google Scholar | DOI

[13] [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] [14] Fathi, A. and Siconolfi, A., Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155(2004), no. 2, 363–388. Google Scholar | DOI

[15] [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] [16] Gomes, D. A., Viscosity solution method and the discrete Aubry-Mather problem. Discrete Contin. Dyn. Syst. 13(2005), no. 1, 103–116. Google Scholar | DOI

[17] [17] Hirsch, M.W., Differential topology. Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994. Google Scholar

[18] [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] [19] Zavidovique, M., Strict sub-solutions and Ma˜ñé potential in discrete weak KAM theory. Comment. Math. Helv. 87(2012), no. 1, 1–39. Google Scholar | DOI

[20] [20] Zavidovique, M., Existence of C1,1 critical subsolutions in discrete weak KAM theory. J. Mod. Dyn. 4(2010), no. 4, 693–714. Google Scholar | DOI

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