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.