Aubry-Mather theory for contact Hamiltonian systems II

Kaizhi Wang; Lin Wang; Jun Yan

doi:10.3934/dcds.2021128

Article Contents

2022, Volume 42, Issue 2: 555-595. Doi: 10.3934/dcds.2021128

This issue Previous Article $ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains Next Article Realizing arbitrary

$d$

-dimensional dynamics by renormalization of

$C^d$

-perturbations of identity

Aubry-Mather theory for contact Hamiltonian systems II

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
3.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^* Corresponding author: Lin Wang
^* Corresponding author: Lin Wang

Received: December 2020

Revised: June 2021

Early access: September 2021

Published: February 2022

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Abstract

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $ H(x,u,p) $ with certain dependence on the contact variable $ u $. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $ \tilde{\mathcal{S}}_s $ consists of strongly static orbits, which coincides with the Aubry set $ \tilde{\mathcal{A}} $ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show $ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of $ H $ on $ u $ fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.

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Mathematics Subject Classification: Primary: 37J50, 35F21; Secondary: 35D40.

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