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August  2019, 39(8): 4345-4358. doi: 10.3934/dcds.2019176

On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem

Kai Zhao and  Wei Cheng ,

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Wei Cheng

Received  April 2018 Revised  January 2019 Published  May 2019

Fund Project: The authors are partly supported by Natural Scientific Foundation of China (Grant No. 11871267, No. 11631006 and No. 11790272).

We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.

Keywords: Hamilton-Jacobi equation, viscosity solution, contact structure.
Mathematics Subject Classification: Primary: 35F21, 49L25; Secondary: 37J50.
Citation: Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176
References:
[1]

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

[2]

P. Cannarsa, Q. Chen and W. Cheng, Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus, Journal of Differential Equations, 2019, arXiv: 1805.10637. doi: 10.1016/j.jde.2019.03.020.  Google Scholar

[3]

P. Cannarsa and W. Cheng, Generalized characteristics and Lax-Oleinik operators: Global theory, Calc. Var. Partial Differential Equations, 56 (2017), Art. 125, 31pp. doi: 10.1007/s00526-017-1219-4.  Google Scholar

[4]

P. Cannarsa, W. Cheng and A. Fathi, On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation, C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180.  doi: 10.1016/j.crma.2016.12.004.  Google Scholar

[5]

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, 2019, arXiv: 1804.03411. Google Scholar

[6]

C. Chen and W. Cheng, Lasry-Lions, Lax-Oleinik and generalized characteristics, Sci. China Math., 59 (2016), 1737-1752.  doi: 10.1007/s11425-016-5143-4.  Google Scholar

[7]

C. Chen, W. Cheng and Q. Zhang, Lasry–Lions approximations for discounted Hamilton–Jacobi equations, J. Differential Equations, 265 (2018), 719-732.  doi: 10.1016/j.jde.2018.03.010.  Google Scholar

[8]

Q. Chen, W. Cheng, H. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations, 2019, arXiv: 1808.06046. Google Scholar

[9]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[10]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[11]

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.  Google Scholar

[12]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988, Translated from the Russian. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[13]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  doi: 10.12775/TMNA.2002.036.  Google Scholar

[14]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  doi: 10.12775/TMNA.2005.034.  Google Scholar

[15]

R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and Hamiltonian Systems, Juliusz Schauder Center for Nonlinear Studies. Nicholas Copernicus University, 1995. Google Scholar

[16]

H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9), 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.  Google Scholar

[17]

H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9), 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.  Google Scholar

[18]

D. E. Varberg, On absolutely continuous functions, Amer. Math. Monthly, 72 (1965), 831–841, https://doi.org/10.2307/2315025. doi: 10.1080/00029890.1965.11970623.  Google Scholar

[19]

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.  Google Scholar

[20]

K. Wang, L. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl. (9), 123 (2019), 167-200.  doi: 10.1016/j.matpur.2018.08.011.  Google Scholar

show all references

References:
[1]

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

[2]

P. Cannarsa, Q. Chen and W. Cheng, Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus, Journal of Differential Equations, 2019, arXiv: 1805.10637. doi: 10.1016/j.jde.2019.03.020.  Google Scholar

[3]

P. Cannarsa and W. Cheng, Generalized characteristics and Lax-Oleinik operators: Global theory, Calc. Var. Partial Differential Equations, 56 (2017), Art. 125, 31pp. doi: 10.1007/s00526-017-1219-4.  Google Scholar

[4]

P. Cannarsa, W. Cheng and A. Fathi, On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation, C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180.  doi: 10.1016/j.crma.2016.12.004.  Google Scholar

[5]

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, 2019, arXiv: 1804.03411. Google Scholar

[6]

C. Chen and W. Cheng, Lasry-Lions, Lax-Oleinik and generalized characteristics, Sci. China Math., 59 (2016), 1737-1752.  doi: 10.1007/s11425-016-5143-4.  Google Scholar

[7]

C. Chen, W. Cheng and Q. Zhang, Lasry–Lions approximations for discounted Hamilton–Jacobi equations, J. Differential Equations, 265 (2018), 719-732.  doi: 10.1016/j.jde.2018.03.010.  Google Scholar

[8]

Q. Chen, W. Cheng, H. Ishii and K. Zhao, Vanishing contact structure problem and convergence of the viscosity solutions, Comm. Partial Differential Equations, 2019, arXiv: 1808.06046. Google Scholar

[9]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.  Google Scholar

[10]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[11]

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.  Google Scholar

[12]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988, Translated from the Russian. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[13]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  doi: 10.12775/TMNA.2002.036.  Google Scholar

[14]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  doi: 10.12775/TMNA.2005.034.  Google Scholar

[15]

R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and Hamiltonian Systems, Juliusz Schauder Center for Nonlinear Studies. Nicholas Copernicus University, 1995. Google Scholar

[16]

H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl. (9), 108 (2017), 125-149.  doi: 10.1016/j.matpur.2016.10.013.  Google Scholar

[17]

H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems, J. Math. Pures Appl. (9), 108 (2017), 261-305.  doi: 10.1016/j.matpur.2016.11.002.  Google Scholar

[18]

D. E. Varberg, On absolutely continuous functions, Amer. Math. Monthly, 72 (1965), 831–841, https://doi.org/10.2307/2315025. doi: 10.1080/00029890.1965.11970623.  Google Scholar

[19]

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.  Google Scholar

[20]

K. Wang, L. Wang and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl. (9), 123 (2019), 167-200.  doi: 10.1016/j.matpur.2018.08.011.  Google Scholar

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