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$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains
Aubry-Mather theory for contact Hamiltonian systems II
1. | School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China |
2. | School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China |
3. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
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.
References:
[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. Bravetti, H. 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. Cannarsa, W. Cheng, L. Jin, K. 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. Chen, W. Cheng, H. 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. Contreras, J. 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. Crandall, H. 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. Davini, A. Fathi, R. 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. Gomes, H. 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. Ishii, H. 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. Ishii, H. 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. Jing, H. 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. Liu, P. 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. Su, L. 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. Wang, L. 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. Wang, L. 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. Wang, L. 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. Wang, L. 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. Wang, J. 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. |
show all references
References:
[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. Bravetti, H. 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. Cannarsa, W. Cheng, L. Jin, K. 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. Chen, W. Cheng, H. 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. Contreras, J. 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. Crandall, H. 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. Davini, A. Fathi, R. 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. Gomes, H. 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. Ishii, H. 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. Ishii, H. 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. Jing, H. 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. Liu, P. 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. Su, L. 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. Wang, L. 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. Wang, L. 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. Wang, L. 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. Wang, L. 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. Wang, J. 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. |
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