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January  2013, 18(1): 57-94. doi: 10.3934/dcdsb.2013.18.57

Weak KAM theory for nonregular commuting Hamiltonians

Andrea Davini 1, and  Maxime Zavidovique 2,

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

Received  May 2011 Revised  June 2012 Published  September 2012

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax--Oleinik semigroups. This is equivalent to the solvability of an associated multi--time Hamilton--Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class $C^{1,1}$ in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.
Keywords: Commuting Hamiltonians, viscosity solutions, Aubry-Mather theory, weak KAM theory, multi-time Hamilton-Jacobi equation..
Mathematics Subject Classification: Primary: 35F21; Secondary: 49L25, 37J5.
Citation: Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
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.  Google Scholar

[2]

O. Alvarez and M. Bardi, "Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations," Mem. Amer. Math. Soc., vol. 204, 2010.  Google Scholar

[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.  Google Scholar

[4]

G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,'' Mathéematiques & Applications, vol. 17. Springer-Verlag, Paris, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

F. H. Clarke, "Optimization and Nonsmooth Analysis,'' Wiley, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

X. Cui, On commuting Tonelli Hamiltonians: time-periodic case, Preprint, (2009). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 13 (1963), 1139-1141.  Google Scholar

[27]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3, (2007), 1037-1055.  Google Scholar

[28]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[29]

A. Fathi, "Weak Kam Theorem in Lagrangian Dynamics,'', Cambridge University Press., ().   Google Scholar

[30]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  Google Scholar

[31]

A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339, doi: 10.1017/S0305004111000661.  Google Scholar

[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.  Google Scholar

[33]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. Imbert and M. Volle, On vectorial Hamilton-Jacobi equations, Control Cybernet, 31 (2002), 493-506.  Google Scholar

[37]

H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135.  Google Scholar

[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.  Google Scholar

[39]

P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1987. Google Scholar

[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.  Google Scholar

[41]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[42]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.  Google Scholar

[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.  Google Scholar

[44]

A. Siconolfi, "Hamilton-Jacobi Equations and Weak KAM Theory," Encyclopedia of Complexity and Systems Science. Springer-Verlag, (2009), 4540-4561. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M. Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808. doi: 10.1088/0951-7715/23/4/002.  Google Scholar

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.  Google Scholar

[2]

O. Alvarez and M. Bardi, "Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations," Mem. Amer. Math. Soc., vol. 204, 2010.  Google Scholar

[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.  Google Scholar

[4]

G. Barles, "Solutions de Viscosité des Équations de Hamilton-Jacobi,'' Mathéematiques & Applications, vol. 17. Springer-Verlag, Paris, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

F. H. Clarke, "Optimization and Nonsmooth Analysis,'' Wiley, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

X. Cui, On commuting Tonelli Hamiltonians: time-periodic case, Preprint, (2009). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 13 (1963), 1139-1141.  Google Scholar

[27]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3, (2007), 1037-1055.  Google Scholar

[28]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[29]

A. Fathi, "Weak Kam Theorem in Lagrangian Dynamics,'', Cambridge University Press., ().   Google Scholar

[30]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  Google Scholar

[31]

A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339, doi: 10.1017/S0305004111000661.  Google Scholar

[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.  Google Scholar

[33]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. Imbert and M. Volle, On vectorial Hamilton-Jacobi equations, Control Cybernet, 31 (2002), 493-506.  Google Scholar

[37]

H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135.  Google Scholar

[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.  Google Scholar

[39]

P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished preprint, 1987. Google Scholar

[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.  Google Scholar

[41]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.  Google Scholar

[42]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.  Google Scholar

[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.  Google Scholar

[44]

A. Siconolfi, "Hamilton-Jacobi Equations and Weak KAM Theory," Encyclopedia of Complexity and Systems Science. Springer-Verlag, (2009), 4540-4561. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

M. Zavidovique, Weak KAM for commuting Hamiltonians, Nonlinearity, 23 (2010), 793-808. doi: 10.1088/0951-7715/23/4/002.  Google Scholar

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