Uniqueness structure of weakly coupled systems of ergodic problems of Hamilton–Jacobi equations
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Here, we address a uniqueness structure of viscosity solutions for ergodic problems of weakly coupled Hamilton–Jacobi systems. In particular, we study comparison principle with respect to generalized Mather measures as a generalization of the result proved by Mitake and Tran, which addressed the case of a single equation. To get the main result, it is important to construct Mather measures effectively. We overcome this difficulty by nonlinear adjoint methods.
KeywordsHamilton–Jacobi equations Weakly coupled systems Viscosity solutions Nonlinear adjoint methods
Mathematics Subject Classification35F21 35A50 37J50
The author would like to thank Professor Hiroyoshi Mitake for his helpful comments and suggestions. The author also would like to appreciate the referees that each of them read this paper very carefully and give him useful comments kindly.
- 3.Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover Publications Inc, New York (1962)Google Scholar
- 8.Le, N.Q., Mitake, H., Tran, H.V.: Dynamical and Geometric Aspects of Hamilton–Jacobi and Linearized Monge–Ampere Equations. VIASM 2016, : Lecture Notes in Mathematics, vol. 2183, p. 2017. Springer, Cham (2016)Google Scholar
- 9.Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge University Press, Cambridge (2003) Google Scholar