Weak KAM theory for discount Hamilton-Jacobi equations and its application

Weak KAM theory for discount Hamilton-Jacobi equations and corresponding discount 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.