Mather problem and viscosity solutions in the stationary setting

2009/03/09 by Diogo A. Gomes, Gomes, Diogo A., Elismar R. Oliveira +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.0903.1594

openalex publication_date 2009/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L:\Rrⁿ× \Rrⁿ× Ω→ \Rr, where Ω is a compact metric space on which \Rrⁿ 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.

Citations

• Action minimizing invariant measures for positive definite Lagrangian systems

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