PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

  • Albert Fathi
  • Antonio Siconolfi


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.


System Theory Viscosity Solution Real Parameter Unique Equation Hamiltonian Flow 
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© Springer-Verlag Berlin/Heidelberg 2005

Authors and Affiliations

  • Albert Fathi
    • 1
  • Antonio Siconolfi
    • 1
  1. 1.Départment de MathématiquesEcole Normale SupérieureLyon Cedex 7France

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