Aubry-Mather measures in the non convex setting

TL;DR

This paper extends the concept of Aubry-Mather measures to non-convex Hamiltonians using the adjoint method, revealing conditions under which invariance is preserved despite potential dissipation.

Contribution

It provides a general construction of Aubry-Mather measures for non-convex Hamiltonians and identifies conditions for their invariance.

Findings

Measures agree with Mather measures in the convex case

Dissipation may occur in non-convex settings

Invariance holds for uniformly quasiconvex Hamiltonians

Abstract

The adjoint method introduced in [Eva] and [Tra] is used, to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the important case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.