Calculus of Variations and Partial Differential Equations

, Volume 22, Issue 2, pp 185–228

PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

  • Albert Fathi
  • Antonio Siconolfi
Article

DOI: 10.1007/s00526-004-0271-z

Cite this article as:
Fathi, A. & Siconolfi, A. Cal Var (2005) 22: 185. doi:10.1007/s00526-004-0271-z

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.

Copyright information

© Springer-Verlag Berlin/Heidelberg 2005

Authors and Affiliations

  • Albert Fathi
    • 1
  • Antonio Siconolfi
    • 1
  1. 1.Départment de MathématiquesEcole Normale SupérieureLyon Cedex 7France
  2. 2.Dipartimento di MatematicaUniversitá degli Studi di Roma “La Sapienza”RomaItaly