Mather problem and viscosity solutions in the stationary setting

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L : Rn ×Rn ×Ω → R, where Ω 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.