2018/01/17 by Wang, Kaizhi, Wang, Lin, Yan, Jun
#35D40 #35F21 #37J50 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1801.05612
This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians H(x,u,p) defined on T^*M×ℝ, satisfying Tonelli conditions with respect to p and 00, where M is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution u-, we prove the existence of the maximal forward weak KAM solution u+. Next, we analyse Aubry set for the contact Hamiltonian system showing that it is the intersection of two Legendrian pseudographs Gu- and Gu+, and that the projection π:T^*M× ℝ→ M induces a bi-Lipschitz homeomorphism π|_A from Aubry set A onto the projected Aubry set A. At last, we introduce the notion of barrier functions and study their interesting properties along calibrated curves. Our analysis is based on a recent method by [43,44].