Some remarks on the continuity equation

We describe some relations between the properties of the Cauchy problem for an ODE and the properties of the Cauchy problem for the associated continuity equation in the class of measures

Young measures, superposition and transport

We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savar\'e on the decomposition of the weak solutions of the transport equation. We also prove, in the context of Mather theory, the equality between Closed measures and Holonomic measures

The dynamics of pseudographs in convex Hamiltonian systems

We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study is inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples

Symplectic aspects of Aubry-Mather theory

We prove that the so-called Aubry and Mane sets introduced by John Mather in Lagrangian dynamics are symplectic invariants. In order to do so, we introduce a barrier in phase space, and propose definitions of Aubry and Mane sets for non-convex Hamiltonian systems. On montre que les ensembles dits d'Aubry et de Mane introduits par John Mather en dynamique Lagrangienne sont des invariants symplectiques. Pour ceci on introduit une barriere sur l'espace des phases, et on definit des ensembles d'Aubry et de Mather pour des systemes Hamiltoniens non convexes

Arnold's Diffusion: from the a priori unstable to the a priori stable case

We expose some selected topics concerning the instability of the action variables in a priori unstable Hamiltonian systems, and outline a new strategy that may allow to apply these methods to a priori stable systems

The Lax-Oleinik semi-group: a Hamiltonian point of view

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.Comment: Proceedings of the Royal Society of Edinburgh, Section: A Mathematics (2012) to appea

Lasry-Lions regularization and a Lemma of Ilmanen

We provide a full self-contained proof of a famous Lemma of Ilmanen. This proof is based on a regularisation procedure similar to Lasry-Lions regularisation

Connecting orbits of time dependent Lagrangian systems

We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one already proposed by Mather. However, its advantage is that it really contains most of the results of Birkhoff and Mather on twist maps