October  2010, 4(4): 693-714. doi: 10.3934/jmd.2010.4.693

Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory

1. 

Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, siteMonod, UMR CNRS 5669, 46, allée d’Italie, 69364 LYON Cedex 07, France

Received  April 2010 Revised  October 2010 Published  January 2011

In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.
Citation: Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
References:
[1]

1-56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988.  Google Scholar

[2]

Adv. Math., 207 (2006), 691-706. doi: 10.1016/j.aim.2006.01.003.  Google Scholar

[3]

J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74.  Google Scholar

[4]

397-420, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007.  Google Scholar

[5]

Ann. Sci. École Norm. Sup. (4), 40 (2007), 445-452.  Google Scholar

[6]

J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[7]

Patrick Bernard, Lasry-Lions regularisation and a Lemma of Ilmanen,, to appear in Rendiconti del Seminario Matematico della Università di Padova., ().   Google Scholar

[8]

Patrick Bernard, Personal communication,, 2009., ().   Google Scholar

[9]

J. Math. Anal. Appl., 260 (2001), 572-601. doi: 10.1006/jmaa.2001.7483.  Google Scholar

[10]

1-21, Adv. Math. Econ., 5, Springer, Tokyo, 2003.  Google Scholar

[11]

preprint, 2000. http://www.cimat.mx/ gonzalo/papers/whj.pdf. Google Scholar

[12]

Springer-Verlag, New York, 1998.  Google Scholar

[13]

Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[14]

Albert Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", preliminary version. , ().   Google Scholar

[15]

Albert Fathi, Personal communication,, 2009., ().   Google Scholar

[16]

Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5.  Google Scholar

[17]

Comm. Pure Appl. Math., 62 (2009), 445-500. doi: 10.1002/cpa.20250.  Google Scholar

[18]

NoDEA, Nonlinear Differential Equations Appl., 14 (2007), 1-27. doi: 10.1007/s00030-007-2047-6.  Google Scholar

[19]

Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.  Google Scholar

[20]

Albert Fathi and Maxime Zavidovique, Insertion of $C^{1,1}$ functions and Ilmanen's lemma,, to appear in Rendiconti del Seminario Matematico della Università di Padova., ().   Google Scholar

[21]

Global variational techniques. Advanced Series in Nonlinear Dynamics, 18, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[22]

Inst. Hautes Études Sci. Publ. Math. No. 70, (1989), 47-101, (1990).  Google Scholar

[23]

193-204, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993.  Google Scholar

[24]

Ergodic Theory Dynam. Systems, 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.  Google Scholar

[25]

Inst. Hautes Études Sci. Publ. Math. No. 63, (1986), 153-204.  Google Scholar

[26]

Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[27]

Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  Google Scholar

[28]

92-186, Lecture Notes in Math., 1589, Springer, Berlin, 1994.  Google Scholar

[29]

Maxime Zavidovique, Strict subsolutions and Mañe potential in discrete weak KAM theory,, to appear in Commentarii Mathematici Helvetici., ().   Google Scholar

show all references

References:
[1]

1-56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988.  Google Scholar

[2]

Adv. Math., 207 (2006), 691-706. doi: 10.1016/j.aim.2006.01.003.  Google Scholar

[3]

J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74.  Google Scholar

[4]

397-420, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007.  Google Scholar

[5]

Ann. Sci. École Norm. Sup. (4), 40 (2007), 445-452.  Google Scholar

[6]

J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[7]

Patrick Bernard, Lasry-Lions regularisation and a Lemma of Ilmanen,, to appear in Rendiconti del Seminario Matematico della Università di Padova., ().   Google Scholar

[8]

Patrick Bernard, Personal communication,, 2009., ().   Google Scholar

[9]

J. Math. Anal. Appl., 260 (2001), 572-601. doi: 10.1006/jmaa.2001.7483.  Google Scholar

[10]

1-21, Adv. Math. Econ., 5, Springer, Tokyo, 2003.  Google Scholar

[11]

preprint, 2000. http://www.cimat.mx/ gonzalo/papers/whj.pdf. Google Scholar

[12]

Springer-Verlag, New York, 1998.  Google Scholar

[13]

Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004.  Google Scholar

[14]

Albert Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", preliminary version. , ().   Google Scholar

[15]

Albert Fathi, Personal communication,, 2009., ().   Google Scholar

[16]

Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5.  Google Scholar

[17]

Comm. Pure Appl. Math., 62 (2009), 445-500. doi: 10.1002/cpa.20250.  Google Scholar

[18]

NoDEA, Nonlinear Differential Equations Appl., 14 (2007), 1-27. doi: 10.1007/s00030-007-2047-6.  Google Scholar

[19]

Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.  Google Scholar

[20]

Albert Fathi and Maxime Zavidovique, Insertion of $C^{1,1}$ functions and Ilmanen's lemma,, to appear in Rendiconti del Seminario Matematico della Università di Padova., ().   Google Scholar

[21]

Global variational techniques. Advanced Series in Nonlinear Dynamics, 18, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[22]

Inst. Hautes Études Sci. Publ. Math. No. 70, (1989), 47-101, (1990).  Google Scholar

[23]

193-204, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993.  Google Scholar

[24]

Ergodic Theory Dynam. Systems, 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.  Google Scholar

[25]

Inst. Hautes Études Sci. Publ. Math. No. 63, (1986), 153-204.  Google Scholar

[26]

Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[27]

Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  Google Scholar

[28]

92-186, Lecture Notes in Math., 1589, Springer, Berlin, 1994.  Google Scholar

[29]

Maxime Zavidovique, Strict subsolutions and Mañe potential in discrete weak KAM theory,, to appear in Commentarii Mathematici Helvetici., ().   Google Scholar

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