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

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

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References:

[1]	G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques & Applications, Springer-Verlag, Paris, 1994.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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