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

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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

Maxime Zavidovique ^1,

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

Abstract
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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.

Keywords: Aubry Mather theory, Hamiltonian and Lagrangian dynamics., Ilmanen's lemma, twist maps, critical subsolutions, Discrete weak KAM theory.

Mathematics Subject Classification: Primary: 37J50; Secondary: 37E40, 37B2.

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]	V. Bangert, Mather sets for twist maps and geodesics on tori, "Dynamics reported, Vol. 1," 1-56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988.
[2]	Patrick Bernard and Boris Buffoni, The Monge problem for supercritical Mañé potentials on compact manifolds, Adv. Math., 207 (2006), 691-706. doi: 10.1016/j.aim.2006.01.003.
[3]	Patrick Bernard and Boris Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74.
[4]	Patrick Bernard and Boris Buffoni, Weak KAM pairs and Monge-Kantorovich duality, "Asymptotic Analysis and Singularities-Elliptic and Parabolic PDEs and Related Problems," 397-420, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007.
[5]	Patrick Bernard, Existence of $C^{1,1}$ critical subsolutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup. (4), 40 (2007), 445-452.
[6]	Patrick Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.
[7]	Patrick Bernard, Lasry-Lions regularisation and a Lemma of Ilmanen, to appear in Rendiconti del Seminario Matematico della Università di Padova.
[8]	Patrick Bernard, Personal communication, 2009.
[9]	Pierre Cardaliaguet, Front propagation problems with nonlocal terms. II, J. Math. Anal. Appl., 260 (2001), 572-601. doi: 10.1006/jmaa.2001.7483.
[10]	Guillaume Carlier, Duality and existence for a class of mass transportation problems and economic applications, "Advances in mathematical economics. Vol. 5," 1-21, Adv. Math. Econ., 5, Springer, Tokyo, 2003.
[11]	Gonzalo Contreras, Renato Iturriaga and Hector Sanchez-Morgado, Weak solutions of the Hamilton-Jacobi equation for time periodic Lagrangians, preprint, 2000. http://www.cimat.mx/ gonzalo/papers/whj.pdf.
[12]	F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," volume 178 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 1998.
[13]	Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004.
[14]	Albert Fathi, "Weak KAM Theorem in Lagrangian Dynamics," preliminary version. http://www.math.u-bordeaux1.fr/ thieulle/Recherche/KamFaible/Publications/Fathi2008_01.pdf
[15]	Albert Fathi, Personal communication, 2009.
[16]	Albert Fathi and Alessio Figalli, Optimal transportation on noncompact manifolds, Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5.
[17]	Albert Fathi, Alessio Figalli and Ludovic Rifford, On the Hausdorff dimension of the Mather quotient, Comm. Pure Appl. Math., 62 (2009), 445-500. doi: 10.1002/cpa.20250.
[18]	A. Fathi and E. Maderna, Weak KAM Theorem on noncompact manifolds, NoDEA, Nonlinear Differential Equations Appl., 14 (2007), 1-27. doi: 10.1007/s00030-007-2047-6.
[19]	Albert Fathi and Antonio Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.
[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.
[21]	Christophe Golé, "Symplectic Twist Maps," Global variational techniques. Advanced Series in Nonlinear Dynamics, 18, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
[22]	Michael-R. Herman, Inégalités "a priori'' pour des tores lagrangiens invariants par des difféomorphismes symplectiques, Inst. Hautes Études Sci. Publ. Math. No. 70, (1989), 47-101, (1990).
[23]	Tom Ilmanen, "The Level-Set Flow on a Manifold," "Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990)," 193-204, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993.
[24]	Daniel Massart, Subsolutions of time-periodic Hamilton-Jacobi equations, Ergodic Theory Dynam. Systems, 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.
[25]	John Mather, A criterion for the nonexistence of invariant circles, Inst. Hautes Études Sci. Publ. Math. No. 63, (1986), 153-204.
[26]	John N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[27]	John N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
[28]	John N. Mather and Giovanni Forni, Action minimizing orbits in Hamiltonian systems, "Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991)," 92-186, Lecture Notes in Math., 1589, Springer, Berlin, 1994.
[29]	Maxime Zavidovique, Strict subsolutions and Mañe potential in discrete weak KAM theory, to appear in Commentarii Mathematici Helvetici.

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References:

[1]	V. Bangert, Mather sets for twist maps and geodesics on tori, "Dynamics reported, Vol. 1," 1-56, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988.
[2]	Patrick Bernard and Boris Buffoni, The Monge problem for supercritical Mañé potentials on compact manifolds, Adv. Math., 207 (2006), 691-706. doi: 10.1016/j.aim.2006.01.003.
[3]	Patrick Bernard and Boris Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74.
[4]	Patrick Bernard and Boris Buffoni, Weak KAM pairs and Monge-Kantorovich duality, "Asymptotic Analysis and Singularities-Elliptic and Parabolic PDEs and Related Problems," 397-420, Adv. Stud. Pure Math., 47-2, Math. Soc. Japan, Tokyo, 2007.
[5]	Patrick Bernard, Existence of $C^{1,1}$ critical subsolutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup. (4), 40 (2007), 445-452.
[6]	Patrick Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.
[7]	Patrick Bernard, Lasry-Lions regularisation and a Lemma of Ilmanen, to appear in Rendiconti del Seminario Matematico della Università di Padova.
[8]	Patrick Bernard, Personal communication, 2009.
[9]	Pierre Cardaliaguet, Front propagation problems with nonlocal terms. II, J. Math. Anal. Appl., 260 (2001), 572-601. doi: 10.1006/jmaa.2001.7483.
[10]	Guillaume Carlier, Duality and existence for a class of mass transportation problems and economic applications, "Advances in mathematical economics. Vol. 5," 1-21, Adv. Math. Econ., 5, Springer, Tokyo, 2003.
[11]	Gonzalo Contreras, Renato Iturriaga and Hector Sanchez-Morgado, Weak solutions of the Hamilton-Jacobi equation for time periodic Lagrangians, preprint, 2000. http://www.cimat.mx/ gonzalo/papers/whj.pdf.
[12]	F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," volume 178 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 1998.
[13]	Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004.
[14]	Albert Fathi, "Weak KAM Theorem in Lagrangian Dynamics," preliminary version. http://www.math.u-bordeaux1.fr/ thieulle/Recherche/KamFaible/Publications/Fathi2008_01.pdf
[15]	Albert Fathi, Personal communication, 2009.
[16]	Albert Fathi and Alessio Figalli, Optimal transportation on noncompact manifolds, Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5.
[17]	Albert Fathi, Alessio Figalli and Ludovic Rifford, On the Hausdorff dimension of the Mather quotient, Comm. Pure Appl. Math., 62 (2009), 445-500. doi: 10.1002/cpa.20250.
[18]	A. Fathi and E. Maderna, Weak KAM Theorem on noncompact manifolds, NoDEA, Nonlinear Differential Equations Appl., 14 (2007), 1-27. doi: 10.1007/s00030-007-2047-6.
[19]	Albert Fathi and Antonio Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. doi: 10.1007/s00222-003-0323-6.
[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.
[21]	Christophe Golé, "Symplectic Twist Maps," Global variational techniques. Advanced Series in Nonlinear Dynamics, 18, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
[22]	Michael-R. Herman, Inégalités "a priori'' pour des tores lagrangiens invariants par des difféomorphismes symplectiques, Inst. Hautes Études Sci. Publ. Math. No. 70, (1989), 47-101, (1990).
[23]	Tom Ilmanen, "The Level-Set Flow on a Manifold," "Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990)," 193-204, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993.
[24]	Daniel Massart, Subsolutions of time-periodic Hamilton-Jacobi equations, Ergodic Theory Dynam. Systems, 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.
[25]	John Mather, A criterion for the nonexistence of invariant circles, Inst. Hautes Études Sci. Publ. Math. No. 63, (1986), 153-204.
[26]	John N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[27]	John N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
[28]	John N. Mather and Giovanni Forni, Action minimizing orbits in Hamiltonian systems, "Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991)," 92-186, Lecture Notes in Math., 1589, Springer, Berlin, 1994.
[29]	Maxime Zavidovique, Strict subsolutions and Mañe potential in discrete weak KAM theory, to appear in Commentarii Mathematici Helvetici.

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