Weak kam theorem on non compact manifolds

By A. Fathi, E. Maderna. 2007

128 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fd6ea8ecbfb9e41bc7b3802afc26ba73ee4fb531

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In Geometry and Topology

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems (2008, background): “The Euler–Lagrange equation generates a flow Lt on TM which is C 1 and complete (see Fathi and Maderna, 2007, Corollary 2.2).”
Lectures on the free period Lagrangian action functional (2013, background): “0(L) is strictly lower than the other two numbers. The values cu(L) and 12 ALBERTO ABBONDANDOLO c0(L) coincide when the fundamental group of Mis Abelian and, more generally, when it is ameanable (see [FM07]). The lowest Man˜´e critical value cu(L) plays a decisive role in the geometry of the action ”
On the Hausdorff dimension of the mather quotient (2007, background): “As it was observed by the first author to generalize Mather’s examples [29], see the announcement [13], a representation formula for δM in term of C 1 critical subsolutions is extremely useful.”
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations (2005, result): “We refer to [Barles] and [Fathi-Maderna] for the study, form a different point of view, of Hamilton-Jacobi equations in the non-compact setting.”
On the Homogenization of the Hamilton-Jacobi Equation (2019, background): “- There exists a unique λ ∈ R for which the following equation admits a viscosity solution v (see [7, 16]): (5) H(x, η(x) + dxv(x)) = λ x ∈ M.”