Weak KAM theorem on non compact manifolds. (English) Zbl 1139.49027

Fathi, Albert; Maderna, Ezequiel

NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 1-2, 1-27 (2007).

Summary: We consider a time independent \(C^2\) 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.

MSC:

49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
70H20	Hamilton-Jacobi equations in mechanics
58D19	Group actions and symmetry properties
49J27	Existence theories for problems in abstract spaces

Keywords:

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