AUBRY-MATHER THEORY FOR CONTACT HAMILTONIAN SYSTEMS II

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 Ss consists of strongly static orbits, which coincides with the Aubry set à in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show Ss $ à 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.