Mather problem and viscosity solutions in the stationary setting

TL;DR

This paper extends the Mather problem to stationary Lagrangians with invariance under group actions, proving the existence of invariant measures and providing estimates for viscosity solutions in a generalized setting.

Contribution

It introduces a generalized Mather problem for stationary Lagrangians invariant under group actions and proves the existence of stationary Mather measures supported in a graph.

Findings

Existence of stationary Mather measures invariant under Euler-Lagrange flow.

Measures are supported in a graph.

Provides estimates for viscosity solutions of Hamilton-Jacobi equations.

Abstract

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians $L:\Rr^n\times \Rr^n\times \Omega\to \Rr$, where $\Omega$ 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.