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Weak kam theorem on non compact manifolds

  • Albert FathiEmail author
  • Ezequiel Maderna
Article

Abstract.

In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.

2000 Mathematics Subject Classification:

49L25 70H20 58D19 

Keywords:

Hamilton Jacobi non compact manifold Lax-Oleinik amenable critical value viscosity solution 

Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Unité de Mathématiques Pures et AppliquésÉcole Normale Supérieure de LyonLyon cedex 07France
  2. 2.Instituto de Matemática y Estadística “Prof. Rafael Laguardia”Universidad de la RepúblicaMontevideoUruguay

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