Properties of action-minimizing sets and weak KAM solutions via Mather's averaging functions

TL;DR

This paper explores the relationship between Mather's averaging functions and the properties of action-minimizing invariant sets and weak KAM solutions in Tonelli systems, revealing connections to system integrability and solution smoothness.

Contribution

It establishes new links between Mather's alpha function features and the geometric and dynamic properties of action-minimizing sets and weak KAM solutions.

Findings

Exposed and extreme points of Mather's alpha function relate to disjointness and graph properties of invariant sets.

Connections between alpha function properties and $C^0$ integrability of systems.

Conditions for the existence of smooth weak KAM solutions based on Mather's functions.

Abstract

We study properties of action-minimizing invariant sets for Tonelli Lagrangian and Hamiltonian systems and weak KAM solutions to the Hamilton-Jacobi equation in terms of Mather's averaging functions. Our principal discovery is that exposed points and extreme points of Mather's alpha function are closely related to disjoint properties and graph properties of the action-minimizing invariant sets, which is also related to $C^0$ integrability of the systems and the existence of smooth weak KAM solutions to the Hamilton-Jacobi equation.