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Weak KAM theory for discounted Hamilton-Jacobi equations and its application. (English) Zbl 1403.35083
Summary: Weak KAM theory for discounted Hamilton-Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then, it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of \(\alpha \)-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of \(\alpha \)-limit points is effectively exploited with properties of the corresponding dynamical systems.

MSC:
35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
37J50 Action-minimizing orbits and measures (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
37K55 Perturbations, KAM for infinite-dimensional Hamiltonian and Lagrangian systems
35D40 Viscosity solutions to PDEs
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