Weak KAM theorem on non compact manifolds

TL;DR

This paper proves the existence of weak KAM solutions for time-independent Hamiltonians on non-compact manifolds using the Lax-Oleinik semigroup, extending classical results to more general settings with symmetries.

Contribution

It provides a new proof of weak KAM solutions on non-compact manifolds and explores the impact of symmetry group amenability on critical values.

Findings

Existence of weak KAM solutions on non-compact manifolds.

Symmetries influence the critical values of the Hamiltonian.

The proof extends classical methods to broader geometric contexts.

Abstract

In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.