Aubry-Mather and weak KAM theories for contact Hamiltonian systems. Part 1: Strictly increasing case

TL;DR

This paper extends Aubry-Mather and weak KAM theories to contact Hamiltonian systems with strictly increasing Hamiltonians, establishing uniqueness, existence, and geometric properties of solutions and Aubry sets.

Contribution

It introduces new results on the uniqueness of backward weak KAM solutions, existence of maximal forward solutions, and geometric analysis of Aubry sets for contact Hamiltonian systems.

Findings

Uniqueness of backward weak KAM solutions established.

Existence of maximal forward weak KAM solutions proved.

Aubry set characterized as intersection of Legendrian pseudographs.

Abstract

This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant \lambda$ for some $\lambda>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry set for the contact Hamiltonian system showing that it is the intersection of two Legendrian pseudographs $G_{u_-}$ and $G_{u_+}$, and that the projection $\pi:T^*M\times \mathbb{R}\to M$ induces a bi-Lipschitz homeomorphism $\pi|_{\tilde{\mathcal{A}}}$ from Aubry set $\tilde{\mathcal{A}}$ onto the projected Aubry set $\mathcal{A}$. At last, we introduce the notion of barrier functions and study their interesting properties along calibrated curves. Our analysis is based on a recent method by [43,44].