Albert Fathi

Weak kam theorem on non compact manifolds

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.

Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions

We consider a continuous coercive Hamiltonian H on the cotangent bundle of the compact connected manifold M which is convex in the momentum. If

Convergence of the solutions of the discounted equation

u_\lambda :M\rightarrow \mathbb {R}

Time functions revisited

uλ:M→R is the viscosity solution of the discounted equation

Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens

\begin{aligned} \lambda u_\lambda (x)+H(x,\mathrm{d}_x u_\lambda )=c(H), \end{aligned}

Optimal transportation on non-compact manifolds

λuλ(x)+H(x,dxuλ)=c(H),where c(H) is the critical value, we prove that