Aubry-Mather theory for contact Hamiltonian systems II

<p style='text-indent:20px;'>In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems <inline-formula><tex-math id="M1">\begin{document}$ H(x,u,p) $\end{document}</tex-math></inline-formula> with certain dependence on the contact variable <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula>. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set <inline-formula><tex-math id="M3">\begin{document}$ \tilde{\mathcal{S}}_s $\end{document}</tex-math></inline-formula> consists of <i>strongly</i> static orbits, which coincides with the Aubry set <inline-formula><tex-math id="M4">\begin{document}$ \tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show <inline-formula><tex-math id="M5">\begin{document}$ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of <inline-formula><tex-math id="M6">\begin{document}$ H $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M7">\begin{document}$ u $\end{document}</tex-math></inline-formula> fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the <i>minimal</i> viscosity solution and <i>non-minimal</i> ones.</p>

Perturbative weak KAM theory

This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.