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Weak KAM theory for nonregular commuting Hamiltonians
1. | Dip. di Matematica, Università di Roma "La Sapienza", P.le Aldo Moro 2, 00185 Roma, Italy |
2. | IMJ, Université Pierre et Marie Curie, Case 247, 4 place Jussieu, F-75252 Paris Cédex 05, France |
References:
[1] |
A. Agrachev and P. W. Y. Lee, Continuity of optimal control costs and its application to weak KAM theory, Calc. Var. Partial Differential Equations, 39 (2010), 213-232.
doi: 10.1007/s00526-010-0308-4. |
[2] |
O. Alvarez and M. Bardi, "Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations," Mem. Amer. Math. Soc., vol. 204, 2010. |
[3] |
L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., 142 (1989), 301-316.
doi: 10.1016/0022-247X(89)90001-2. |
[4] |
G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,'' Mathéematiques & Applications, vol. 17. Springer-Verlag, Paris, 1994. |
[5] |
G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466.
doi: 10.1007/s00526-007-0097-6. |
[6] |
G. Barles and I. Mitake, A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations, Comm. Partial Differential Equations, 37 (2012), 136-168.
doi: 10.1080/03605302.2011.553645. |
[7] |
G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939 |
[8] |
G. Barles and A. Tourin, Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations, Indiana Univ. Math. J., 50 (2001), 1523-1544. |
[9] |
P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup. 4, 40 (2007), 445-452. |
[10] |
P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420. |
[11] |
G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems. An Introduction,'' Oxford Lecture Series in Mathematics and its Applications, 15. The Clarendon Press, Oxford University Press, New York, 1998. |
[12] |
F. Camilli, A. Cesaroni and A. Siconolfi, Randomly perturbed dynamical systems and Aubry-Mather theory, Int. J.Dyn. Syst. Differ. Equ., 2 (2009), 139-169. |
[13] |
P. Cannarsa and C. 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] |
P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $\mathbb R^2/\mathbb Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856.
doi: 10.1016/j.anihpc.2009.11.015. |
[15] |
F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[16] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,'' Wiley, New York, 1983. |
[17] |
F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98.
doi: 10.1090/S0002-9947-1985-0779053-3. |
[18] |
G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[19] |
X. Cui, On commuting Tonelli Hamiltonians: time-periodic case, Preprint, (2009). |
[20] |
X. Cui and J. Li, On commuting Tonelli Hamiltonians: autonomous case, J. Diff. Equations, 250 (2011), 4104-4123.
doi: 10.1016/j.jde.2011.01.020. |
[21] |
A. Davini, Bolza problems with discontinuous Lagrangians and Lipschitz continuity of the value function, SIAM J. Control Optim., 46 (2007), 1897-1921.
doi: 10.1137/060654311. |
[22] |
A. Davini and A. Siconolfi, A generalized dynamical approachto the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[23] |
A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the $1$-dimensional case, Math. Ann., 345 (2009), 749-782.
doi: 10.1007/s00208-009-0372-2. |
[24] |
A. Davini and A. Siconolfi, Metric techniques for convex stationary ergodic Hamiltonians, Calc. Var. Partial Differential Equations, 40 (2011), 391-421.
doi: 10.1007/s00526-010-0345-z. |
[25] |
A. Davini and A. Siconolfi, Weak KAM Theory topics in the stationary ergodic setting, Calc. Var. Partial Differential Equations, 44 (2012), 319-350.
doi: 10.1007/s00526-011-0436-5. |
[26] |
R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 13 (1963), 1139-1141. |
[27] |
M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3, (2007), 1037-1055. |
[28] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998. |
[29] |
A. Fathi, "Weak Kam Theorem in Lagrangian Dynamics,'' Cambridge University Press. |
[30] |
A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. |
[31] |
A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339,
doi: 10.1017/S0305004111000661. |
[32] |
A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for continuous convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[33] |
M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347.
doi: 10.1007/BF01388806. |
[34] |
N. Ichihara and H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians, Comm. Partial Differential Equations, 33 (2008), 784-807.
doi: 10.1080/03605300701257427. |
[35] |
N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.
doi: 10.1007/s00205-008-0170-0. |
[36] |
C. Imbert and M. Volle, On vectorial Hamilton-Jacobi equations, Control Cybernet, 31 (2002), 493-506. |
[37] |
H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135. |
[38] |
H. Ishii and H. Mitake, Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians, Indiana Univ. Math. J., 56 (2007), 2159-2183.
doi: 10.1512/iumj.2007.56.3048. |
[39] |
P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1987. |
[40] |
P.-L. Lions and J.-C. Rochet, Hopf formula and multitime Hamilton-Jacobi equations, Proc. Amer. Math. Soc., 96 (1986), 79-84.
doi: 10.1090/S0002-9939-1986-0813815-5. |
[41] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[42] |
J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[43] |
M. Motta and F. Rampazzo, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J., 55 (2006), 1573-1614.
doi: 10.1512/iumj.2006.55.2760. |
[44] |
A. Siconolfi, "Hamilton-Jacobi Equations and Weak KAM Theory," Encyclopedia of Complexity and Systems Science. Springer-Verlag, (2009), 4540-4561. |
[45] |
A. Siconolfi and G. Terrone, A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics, Nonlinearity, 20 (2007), 1077-1093.
doi: 10.1088/0951-7715/20/5/002. |
[46] |
A. Sorrentino, On the integrability of Tonelli Hamiltonians, Trans. Amer. Math. Soc., 363 (2011), 5071-5089.
doi: 10.1090/S0002-9947-2011-05492-9. |
[47] |
M. Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808.
doi: 10.1088/0951-7715/23/4/002. |
show all references
References:
[1] |
A. Agrachev and P. W. Y. Lee, Continuity of optimal control costs and its application to weak KAM theory, Calc. Var. Partial Differential Equations, 39 (2010), 213-232.
doi: 10.1007/s00526-010-0308-4. |
[2] |
O. Alvarez and M. Bardi, "Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations," Mem. Amer. Math. Soc., vol. 204, 2010. |
[3] |
L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., 142 (1989), 301-316.
doi: 10.1016/0022-247X(89)90001-2. |
[4] |
G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,'' Mathéematiques & Applications, vol. 17. Springer-Verlag, Paris, 1994. |
[5] |
G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466.
doi: 10.1007/s00526-007-0097-6. |
[6] |
G. Barles and I. Mitake, A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations, Comm. Partial Differential Equations, 37 (2012), 136-168.
doi: 10.1080/03605302.2011.553645. |
[7] |
G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939 |
[8] |
G. Barles and A. Tourin, Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations, Indiana Univ. Math. J., 50 (2001), 1523-1544. |
[9] |
P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup. 4, 40 (2007), 445-452. |
[10] |
P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420. |
[11] |
G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems. An Introduction,'' Oxford Lecture Series in Mathematics and its Applications, 15. The Clarendon Press, Oxford University Press, New York, 1998. |
[12] |
F. Camilli, A. Cesaroni and A. Siconolfi, Randomly perturbed dynamical systems and Aubry-Mather theory, Int. J.Dyn. Syst. Differ. Equ., 2 (2009), 139-169. |
[13] |
P. Cannarsa and C. 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] |
P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $\mathbb R^2/\mathbb Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856.
doi: 10.1016/j.anihpc.2009.11.015. |
[15] |
F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[16] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,'' Wiley, New York, 1983. |
[17] |
F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98.
doi: 10.1090/S0002-9947-1985-0779053-3. |
[18] |
G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[19] |
X. Cui, On commuting Tonelli Hamiltonians: time-periodic case, Preprint, (2009). |
[20] |
X. Cui and J. Li, On commuting Tonelli Hamiltonians: autonomous case, J. Diff. Equations, 250 (2011), 4104-4123.
doi: 10.1016/j.jde.2011.01.020. |
[21] |
A. Davini, Bolza problems with discontinuous Lagrangians and Lipschitz continuity of the value function, SIAM J. Control Optim., 46 (2007), 1897-1921.
doi: 10.1137/060654311. |
[22] |
A. Davini and A. Siconolfi, A generalized dynamical approachto the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[23] |
A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the $1$-dimensional case, Math. Ann., 345 (2009), 749-782.
doi: 10.1007/s00208-009-0372-2. |
[24] |
A. Davini and A. Siconolfi, Metric techniques for convex stationary ergodic Hamiltonians, Calc. Var. Partial Differential Equations, 40 (2011), 391-421.
doi: 10.1007/s00526-010-0345-z. |
[25] |
A. Davini and A. Siconolfi, Weak KAM Theory topics in the stationary ergodic setting, Calc. Var. Partial Differential Equations, 44 (2012), 319-350.
doi: 10.1007/s00526-011-0436-5. |
[26] |
R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 13 (1963), 1139-1141. |
[27] |
M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3, (2007), 1037-1055. |
[28] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998. |
[29] |
A. Fathi, "Weak Kam Theorem in Lagrangian Dynamics,'' Cambridge University Press. |
[30] |
A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. |
[31] |
A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339,
doi: 10.1017/S0305004111000661. |
[32] |
A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for continuous convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[33] |
M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347.
doi: 10.1007/BF01388806. |
[34] |
N. Ichihara and H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians, Comm. Partial Differential Equations, 33 (2008), 784-807.
doi: 10.1080/03605300701257427. |
[35] |
N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.
doi: 10.1007/s00205-008-0170-0. |
[36] |
C. Imbert and M. Volle, On vectorial Hamilton-Jacobi equations, Control Cybernet, 31 (2002), 493-506. |
[37] |
H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135. |
[38] |
H. Ishii and H. Mitake, Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians, Indiana Univ. Math. J., 56 (2007), 2159-2183.
doi: 10.1512/iumj.2007.56.3048. |
[39] |
P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1987. |
[40] |
P.-L. Lions and J.-C. Rochet, Hopf formula and multitime Hamilton-Jacobi equations, Proc. Amer. Math. Soc., 96 (1986), 79-84.
doi: 10.1090/S0002-9939-1986-0813815-5. |
[41] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[42] |
J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[43] |
M. Motta and F. Rampazzo, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana Univ. Math. J., 55 (2006), 1573-1614.
doi: 10.1512/iumj.2006.55.2760. |
[44] |
A. Siconolfi, "Hamilton-Jacobi Equations and Weak KAM Theory," Encyclopedia of Complexity and Systems Science. Springer-Verlag, (2009), 4540-4561. |
[45] |
A. Siconolfi and G. Terrone, A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics, Nonlinearity, 20 (2007), 1077-1093.
doi: 10.1088/0951-7715/20/5/002. |
[46] |
A. Sorrentino, On the integrability of Tonelli Hamiltonians, Trans. Amer. Math. Soc., 363 (2011), 5071-5089.
doi: 10.1090/S0002-9947-2011-05492-9. |
[47] |
M. Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808.
doi: 10.1088/0951-7715/23/4/002. |
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