Weak KAM theorem on non compact manifolds

Abstract

In this paper, we consider a time independent $C^2$ 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.

arXiv admin note: text overlap with arXiv:1004.0086 by other authors

References in corpus (1)

• On the Homogenization of the Hamilton-Jacobi Equation

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• A Warped Product Splitting Theorem Through Weak KAM Theory
• Mather problem and viscosity solutions in the stationary setting
• Weak KAM Theorem for a Class of Infinite-Dimensional Lagrangian systems
• Invariance of global solutions of the Hamilton-Jacobi equation