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On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem
Department of Mathematics, Nanjing University, Nanjing 210093, China |
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.
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. |
show all references
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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