PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

Fathi, Albert; Siconolfi, Antonio
February 2005
Calculus of Variations & Partial Differential Equations;Feb2005, Vol. 22 Issue 2, p185
Academic Journal
We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus) just coercive, continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equationswithareal parameter, and in particular on the unique equation of the family, corresponding to the so-called critical valuea=c, for which there is a viscosity solution on. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.


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