2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222712In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians $L:\Rr^n\times \Rr^n\times Ω\to \Rr$, where $Ω$ is a compact metric space on which $\Rr^n$ 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.Analysis of PDEsDynamical Systemstext