Advanced Search
Article Contents
Article Contents

Aubry-Mather theory for contact Hamiltonian systems II

  • * Corresponding author: Lin Wang

    * Corresponding author: Lin Wang 
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $ H(x,u,p) $ with certain dependence on the contact variable $ u $. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $ \tilde{\mathcal{S}}_s $ consists of strongly static orbits, which coincides with the Aubry set $ \tilde{\mathcal{A}} $ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show $ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of $ H $ on $ u $ fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.

    Mathematics Subject Classification: Primary: 37J50, 35F21; Secondary: 35D40.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] V. Arnold, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.
    [2] P. Bernard, Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. École Norm. Sup., 40 (2007), 445-452.  doi: 10.1016/j.ansens.2007.01.004.
    [3] P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  doi: 10.1215/S0012-7094-07-13631-7.
    [4] P. 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.
    [5] A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12 pp. doi: 10.3390/e19100535.
    [6] A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Meth. Mod. Phys., 16 (2019), 1940003.  doi: 10.1142/S0219887819400036.
    [7] A. BravettiH. Cruz and D. Tapias, Contact Hamiltonian mechanics, Ann. Physics, 376 (2017), 17-39.  doi: 10.1016/j.aop.2016.11.003.
    [8] P. Cannarsa, W. Cheng, K. Wang and J. Yan, Herglotz' generalized variational principle and contact type Hamilton-Jacobi equations, Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., Springer, Cham, 32 (2019), 39–67.
    [9] P. CannarsaW. ChengL. JinK. Wang and J. Yan, Herglotz' variational principle and Lax-Oleinik evolution, J. Math. Pures Appl., 141 (2020), 99-136.  doi: 10.1016/j.matpur.2020.07.002.
    [10] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, In Nonlinear Differential Equations and Their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.
    [11] Q. Chen, Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function, Adv. Calc. Var., Published online.
    [12] Q. ChenW. ChengH. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations, 44 (2019), 801-836.  doi: 10.1080/03605302.2019.1608561.
    [13] G. ContrerasJ. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155-196.  doi: 10.1007/BF01233390.
    [14] M. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.
    [15] G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22nd Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.
    [16] M. 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.
    [17] M. de León and C. Sardón, Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems, J. Phys. A, 50 (2017), 255205.  doi: 10.1088/1751-8121/aa711d.
    [18] M. de León and M. L. Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geome. Phys., 153 (2020), 103651.  doi: 10.1016/j.geomphys.2020.103651.
    [19] A. DaviniA. FathiR. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation: Convergence of the discounted solutions, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6.
    [20] A. Davini and L. Wang, On the vanishing discount problems from the negative direction, Discrete Contin. Dyn. Syst., 41 (2021), 2377-2389.  doi: 10.3934/dcds.2020368.
    [21] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, preliminary version 10, Lyon, unpublished (2008).
    [22] A. Fathi and A. 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.
    [23] D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1 (2008), 291-307.  doi: 10.1515/ACV.2008.012.
    [24] D. GomesH. Mitake and H. Tran, The selection problem for discounted Hamilton-Jacobi equations: Some non-convex cases, J. Math. Soc. Jpn., 70 (2018), 345-364.  doi: 10.2969/jmsj/07017534.
    [25] G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930.
    [26] R. Iturriaga and H. Sanchez-Morgado, Limit of the infinite horizon discounted Hamilton-Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 623-635.  doi: 10.3934/dcdsb.2011.15.623.
    [27] H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl., 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.
    [28] H. IshiiH. Mitake and H. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl., 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.
    [29] W. JingH. Mitake and H. Tran, Generalized ergodic problems: Existence and uniqueness structures of solutions, J. Differential Equations, 268 (2020), 2886-2909.  doi: 10.1016/j.jde.2019.09.046.
    [30] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi Equations, preprint.
    [31] Q. LiuP. Torres and C. Wang, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.  doi: 10.1016/j.aop.2018.04.035.
    [32] S. Marò and A. Sorrentino, Aubry-Mather theory for conformally symplectic systems, Commun. Math. Phys., 354 (2017), 775-808.  doi: 10.1007/s00220-017-2900-3.
    [33] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.
    [34] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.
    [35] R. Mañé, Lagrangain flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Math., 28 (1997), 141-153.  doi: 10.1007/BF01233389.
    [36] H. Mitake and K. Soga, Weak KAM theory for discounted Hamilton-Jacobi equations and its application, Calc. Var. Partial Differential Equations, 57 (2018), 57-78.  doi: 10.1007/s00526-018-1359-1.
    [37] H. Mitake and H. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Adv. Math., 306 (2017), 684-703.  doi: 10.1016/j.aim.2016.10.032.
    [38] K. Soga, Selection problems of $\mathbb{Z}^2$ -periodic entropy solutions and viscosity solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 30pp. doi: 10.1007/s00526-017-1208-7.
    [39] X. SuL. Wang and J. Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unkown functions, Discrete Contin. Dyn. Syst., 36 (2016), 6487-6522.  doi: 10.3934/dcds.2016080.
    [40] M. L. Valcázar and M. de León, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 102902.  doi: 10.1063/1.5096475.
    [41] K. WangL. Wang and J. Yan, Implicit variational principle for contact Hamiltonian systems, Nonlinearity, 30 (2017), 492-515.  doi: 10.1088/1361-6544/30/2/492.
    [42] K. WangL. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.  doi: 10.1016/j.matpur.2018.08.011.
    [43] K. WangL. Wang and J. Yan, Aubry-Mather theory for contact Hamiltonian systems, Commun. Math. Phys., 366 (2019), 981-1023.  doi: 10.1007/s00220-019-03362-2.
    [44] K. WangL. Wang and J. Yan, Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions, J. Differential Equations, 286 (2021), 411-432.  doi: 10.1016/j.jde.2021.03.030.
    [45] Y. Wang and J. Yan, A variational principle for contact Hamiltonian systems, J. Differential Equations, 267 (2019), 4047-4088.  doi: 10.1016/j.jde.2019.04.031.
    [46] Y. WangJ. Yan and J. Zhang, Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations, Arch. Rational Mech. Anal., 241 (2021), 885-902.  doi: 10.1007/s00205-021-01667-y.
    [47] K. Zhao and W. Cheng, On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem, Discrete Contin. Dyn. Syst., 39 (2019), 4345-4358.  doi: 10.3934/dcds.2019176.
  • 加载中

Article Metrics

HTML views(1964) PDF downloads(261) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint