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- W3199715351 abstract "<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}}_ssubsetneqqtilde{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>" @default.
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- W3199715351 date "2021-01-01" @default.
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- W3199715351 title "Aubry-Mather theory for contact Hamiltonian systems II" @default.
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- W3199715351 doi "https://doi.org/10.3934/dcds.2021128" @default.
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