Mather problem and viscosity solutions in the stationary setting

Abstract

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L : Rn × Rn ×! R, where is a compact metric space on which Rn acts through an action which leaves L invariant. This setting allow us to generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians. Our main result is the existence of stationary Mather measures invariant under the Euler-Lagrange flow which are supported in a graph. We also obtain several estimates for viscosity solutions of Hamilton-Jacobi equations for the discounted cost infinite horizon problem.