  The Weak KAM theory was developed by Fathi in order to study the dynamics of
convex Hamiltonian systems. It somehow makes a bridge between viscosity
solutions of the Hamilton-Jacobi equation and Mather invariant sets of
Hamiltonian systems, although this was fully understood only a posteriori.
These theories converge under the hypothesis of convexity, and the richness of
applications mostly comes from this remarkable convergence. In the present
course, we provide an elementary exposition of some of the basic concepts of
weak KAM theory. In a companion lecture, Albert Fathi exposes the aspects of
his theory which are more directly related to viscosity solutions. Here on the
contrary, we focus on dynamical applications, even if we also discuss some
viscosity aspects to underline the connections with Fathi's lecture. The
fundamental reference on Weak KAM theory is the still unpublished book of
Albert Fathi \textit{Weak KAM theorem in Lagrangian dynamics}. Although we do
not offer new results, our exposition is original in several aspects. We only
work with the Hamiltonian and do not rely on the Lagrangian, even if some
proofs are directly inspired from the classical Lagrangian proofs. This
approach is made easier by the choice of a somewhat specific setting. We work
on $\Rm^d$ and make uniform hypotheses on the Hamiltonian. This allows us to
replace some compactness arguments by explicit estimates. For the most
interesting dynamical applications however, the compactness of the
configuration space remains a useful hypothesis and we retrieve it by
considering periodic (in space) Hamiltonians. Our exposition is centered on the
Cauchy problem for the Hamilton-Jacobi equation and the Lax-Oleinik evolution
operators associated to it. Dynamical applications are reached by considering
fixed points of these evolution operators, the Weak KAM solutions. The
evolution operators can also be used for their regularizing properties, this
opens a second way to dynamical applications.
