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Abstract:
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still unpublished book of Albert Fathi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.
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