PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

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Abstract.

We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus \(\mathbb{T}^N\) ) 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 \(\mathbb{T}^N\) . We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.

Received: 23 November 2003, Accepted: 3 March 2004, Published online: 12 May 2004