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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 |
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] | |
[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. , ().
|
[15] | |
[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., ().
|
show all references
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] | |
[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. , ().
|
[15] | |
[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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