PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

Abstract

We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus T-N) just coercive. continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations H(x, Du) = a with a real parameter, and in particular on the unique equation of the family. corresponding to the so-called critical value a = c, for which there is a viscosity solution on T-N. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians