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

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

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.
Citation: Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and 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.

[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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