2015/02/22 by Albert Fathi, Ezequiel Maderna
Mathematics · #math.DS #msc:49L25 #msc:70H20 #msc:58D19
paper · pdf · doi:10.1007/s00030-007-2047-6
published as Nonlinear Differential Equations and Applications NoDEA, October 2007, Volume 14, Issue 1-2, pp 1-27 · arXiv admin note: text overlap with arXiv:1004.0086 by other authors
arxiv created 2015/02/22 · arxiv updated 2015/02/24
In this paper, we consider a time independent C2 Hamiltonian, sa\-tisfying 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.