The Lax-Oleinik semi-group on graphs

TL;DR

This paper studies the Lax-Oleinik semi-group on graphs, establishing the existence, uniqueness, and convergence of weak KAM solutions, and linking them to viscosity solutions, with implications for Hamiltonians of eikonal type.

Contribution

It introduces a framework for weak KAM solutions on graphs, proves their long-term convergence, and characterizes their relation to viscosity solutions for specific Hamiltonians.

Findings

Weak KAM solutions are fixed points of the Lax-Oleinik semi-group.

The Aubry set characterizes the uniqueness of solutions.

The Lax-Oleinik semi-group converges over time to a solution.

Abstract

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. Weak KAM solutions are viscosity solutions, and in the case of Hamiltonians called of eikonal type in [CS], we prove that the converse holds.