Time-periodic Evans approach to weak KAM theory

TL;DR

This paper extends Evans' approach to weak KAM theory to time-periodic systems, establishing existence and smoothness of solutions, and analyzing their limiting behavior in the context of mean field games.

Contribution

It introduces a time-periodic Evans approach, proves smoothness of solutions, and analyzes the limiting behavior of related quantities in weak KAM theory.

Findings

Existence of smooth solutions for the time-periodic Evans problem.

Smoothness of the effective Lagrangian and Hamiltonian.

Analysis of the limiting behavior of Mather measures and minimizers.

Abstract

We study the time-periodic version of Evans approach to weak KAM theory. Evans minimization problem is equivalent to a first oder mean field game system. For the mechanical Hamiltonian we prove the existence of smooth solutions. We introduce the corresponding effective Lagrangian and Hamiltonian and prove that they are smooth. We also consider the limiting behavior of the effective Lagrangian and Hamiltonian, Mather measures and minimizers.