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arXiv · 2107.07088

Aubry-Mather theory for contact Hamiltonian systems II

Kaizhi Wang·Lin Wang·Jun Yan
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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 Ma\~{n}\'{e} 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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Kaizhi Wang, Lin Wang, Jun Yan. 2021-07-15. Aubry-Mather theory for contact Hamiltonian systems II. https://arxiv.org/abs/2107.07088

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