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Aubry-Mather theory for contact Hamiltonian systems II

  • * Corresponding author: Lin Wang

    * Corresponding author: Lin Wang 
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  • 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.

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


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