# Mather problem and viscosity solutions in the stationary setting - Mathematics > Analysis of PDEs

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Abstract: In this paper we discuss the Mather problem for stationary Lagrangians, thatis Lagrangians $L:\Rr^n\times \Rr^n\times \Omega\to \Rr$, where $\Omega$ is acompact metric space on which $\Rr^n$ acts through an action which leaves $L$invariant.

This setting allow us to generalize the standard Mather problem forquasi-periodic and almost-periodic Lagrangians.

Our main result is theexistence of stationary Mather measures invariant under the Euler-Lagrange flowwhich are supported in a graph.

We also obtain several estimates for viscositysolutions of Hamilton-Jacobi equations for the discounted cost infinite horizonproblem.

Author: ** Diogo A.Gomes, Elismar R.Oliveira**

Source: https://arxiv.org/