Stability and weak KAM solutions of contact Hamilton-Jacobi equation

Xu, Yang; Yan, Jun; Zhao, Kai

doi:10.1090/proc/16611

Stability and weak KAM solutions of contact Hamilton-Jacobi equation
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by Yang Xu, Jun Yan and Kai Zhao;

Proc. Amer. Math. Soc. 152 (2024), 725-738

DOI: https://doi.org/10.1090/proc/16611

Published electronically: November 29, 2023

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Abstract:

We are concerned with the stability of viscosity solutions to contact Hamilton-Jacobi equation \begin{align*} H(x,\partial _x u(x),u(x))=0, \quad x\in M, \end{align*} where $H=H(x,p,u)$ satisfies Tonelli conditions. We study the relationship between Lyapunov stability of viscosity solutions and the structure of the set of weak KAM solutions to the contact Hamilton-Jacobi equation.

References

Piermarco Cannarsa, Wei Cheng, Liang Jin, Kaizhi Wang, and Jun Yan, Herglotz’ variational principle and Lax-Oleinik evolution, J. Math. Pures Appl. (9) 141 (2020), 99–136. MR 4134452, DOI 10.1016/j.matpur.2020.07.002
Piermarco Cannarsa, Wei Cheng, Kaizhi Wang, and Jun Yan, Herglotz’ generalized variational principle and contact type Hamilton-Jacobi equations, Trends in control theory and partial differential equations, Springer INdAM Ser., vol. 32, Springer, Cham, 2019, pp. 39–67. MR 3965251
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
Albert Fathi, Weak KAM theorem in Lagrangian dynamics, Cambridge University Press, Cambridge, 2008.
Hitoshi Ishii, Kaizhi Wang, Lin Wang, and Jun Yan, Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown, Comm. Partial Differential Equations 47 (2022), no. 2, 417–452. MR 4378613, DOI 10.1080/03605302.2021.1983598
Liang Jin, Jun Yan, and Kai Zhao, Nonlinear semigroup approach to the Hamilton-Jacobi equation—a toy model, Minimax Theory Appl. 8 (2023), no. 1, 61–84. MR 4567767
P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, Unpublished work, 1987.
Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
Ricardo Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 (1996), no. 2, 273–310. MR 1384478, DOI 10.1088/0951-7715/9/2/002
Kaizhi Wang, Lin Wang, and Jun Yan, Implicit variational principle for contact Hamiltonian systems, Nonlinearity 30 (2017), no. 2, 492–515. MR 3604353, DOI 10.1088/1361-6544/30/2/492
Kaizhi Wang, Lin Wang, and Jun Yan, Aubry-Mather theory for contact Hamiltonian systems, Comm. Math. Phys. 366 (2019), no. 3, 981–1023. MR 3927084, DOI 10.1007/s00220-019-03362-2
Kaizhi Wang, Lin Wang, and Jun Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl. (9) 123 (2019), 167–200 (English, with English and French summaries). MR 3912650, DOI 10.1016/j.matpur.2018.08.011
Kaizhi Wang, Lin Wang, and Jun Yan, Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions, J. Differential Equations 286 (2021), 411–432. MR 4234811, DOI 10.1016/j.jde.2021.03.030
Kaizhi Wang and Jun Yan, Ergodic problems for contact Hamilton-Jacobi equations, arXiv:2107.11554.
Kaizhi Wang, Jun Yan, and Kai Zhao, Finite-time convergence of solutions of Hamilton-Jacobi equations, Proc. Amer. Math. Soc. 150 (2022), no. 3, 1187–1196. MR 4375713, DOI 10.1090/proc/15736