Eksistensi dan Ketunggalan Solusi Viscosity Masalah Ergodik
Keywords:
ergodic problem, Hamilton-Jacobi equation, viscosity solutionAbstract
The study of viscosity solution to Hamilton-Jacobi equation was attracted researchers in the last decades. The study focused on completing the question regarding existence, uniqueness, and regularity of the solution. In this research, we covered the existence of viscosity solutions to the ergodic problem or Hamilton-Jacobi equation of contact type in the n-dimension with some general assumptions. The viscosity solutions obtained are not unique in general. A prototype of the Hamiltonian class was investigated and analyzed to show the problem the uniqueness of the viscosity solution. We used the method in Jing (2020:6) to obtain the existence and uniqueness of the ergodic problem.
References
Barles, G., Souganidis, P.E. (2000). On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31, no. 4, 925-939.
Crandall, M.G., Evans, L.C., Lions, P.-L. (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282, 487-502.
Crandall, M.G., Lions, P.-L. (1983). Viscosity Solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277, 1-42.
Evans, L.C. (2010). Partial Differential Equations Second Edition, American Mathematical Society.
Ishii, H. (1987). Perron's method for Hamilton-Jacobi equations. Duke Math. Journal 55, no. 2, 369-384.
Jing, W., Mitake, H., Tran, H.V. (2020). Generalized ergodic problems: Existence and uniqueness structures of solutions. Journal of Differential Equations, 268, 2886-2909.
Lions, P.L., Papanicolaou, G.C., Varadhan, S.R.S. (1987). Homogenization of Hamilton-Jacobi equation. Unpublished preprint.
Le, N.Q., Mitake, H., Tran, H.V. (2016). Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampere Equations. Lecture Notes in Mathematics 2183, Springer.
Su, X., Wang, L., Yan, J. (2016). Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions. Discrete and Continuous Dynamical System 36(11):6487-6522.
Wang, K., Wang, L., Yan, J. (2019). Variational principle for contact Hamiltonian systems and its applications. Journal de Mathèmatiques Pures et Appliquèes, 167-200.
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