Given a minimizing measure of a Tonelli Hamiltonian or a symplectic positive twist map of the double dimensional annulus, we will explain the link connecting the Lyapunov exponents and the mean angle between the Oseledet's splittings for the measure. To do that, we use the so-called Green bundles.

We consider Hamiltonians H convex in the first n moment variables and concave in the other m, e.g., the difference of two usual convex and coercive Hamiltonians, the former in n and the latter in m variables. At least formally, one can associate to it a variational problem where one seeks trajectories whose first n components minimize and the last m maximize a convex-concave Lagrangian. We show that the theory of differential games allows to make this link rigorous, at least in part. We also define as critical value of H the infimum of the constants c such that the stationary Hamilton-Jacobi equation with right hand side c has a (viscosity) subsolution. We prove that some of the properties of the critical value well-known in the convex case still hold for a large class of non-convex Hamiltonians. Finally we prove the existence and a formula for the critical value of some convex-concave Hamiltonians.

We consider a recent formulation of weak KAM theory proposed by Evans. We point out that all the corresponding computations can be explicitly done not only for the classical integrable case, but also for 1-dimensional mechanical Hamiltonian systems. In such a setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamical interpretation. We finally discuss some consequences for the multidimensional case. Joint work with Franco Cardin and Massimiliano Guzzo.

We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. We present a new and simple proof of the rate of convergence of the approximations based on the adjoint method recently introduced by Evans. This is a joint work with Diogo Gomes and Hung V.Tran.

An important problem in graph theory is to detect the shortest paths connecting the vertices of a graph to a prescribed target vertex. Here we study a generalization: to find the shortest path connecting any point of a graph, and not only a vertex, to the target. Our approach is based on the study of Eikonal equations and on the corresponding theory of viscosity solutions on topological graphs.

I will consider the evolution by mean curvature in a heterogeneous medium, modeled by a periodic forcing term. I will discuss two related problems: the existence of a homogenization limit, when the dimension of the periodicity cell tends to zero, and the long time behaviour of the evolution, in particular the existence of travelling waves solutions. Joint work with Guy Barles and Matteo Novaga.

We derive and apply a logarithmic Sobolev inequality with Lipschitz constants on the n-dimensional Euclidean space. This inequality is induced by Hamilton-Jacobi equations. As applications, we provide a representation of Lipschitz constants by the entropy and hypercontractivity of solutions of some functional equations.

We study the stochastic version of the Evans-Aronsson problem. We prove existence of solutions and establish that the corresponding effective Lagrangian and Hamiltonian are smooth both for the case of mechanical Hamiltonians in any dimension or for general Hamiltonians in dimension 2. Finally, we study the limiting behavior. This is a joint work with R. Iturriaga and H. Sanchez-Morgado.

We will consider weak KAM solutions of supercritical or critical Hamilton-Jacobi equations related to the classical N-body problem in some Euclidean space of dimension at least two. We will show that semi-static curves defined on unbounded intervals must have fixed center of mass. This will apply to calibrated curves of weak KAM solutions, allowing to deduce that all these solutions are invariant under translations in the space of configurations. We will also prove the existence of non-invariant solutions for super-critical equations.

Suppose that we have two one-periodic Tonelli Hamiltonians H,G defined on the cotangent space of a manifold M, and suppose that we can iterate their time-one maps in any order. What kind of trajectories can be constructed in this way?

We consider real-valued continuous functions from an open set of an Euclidean n-dimensional space with epigraph possessing (locally) positive reach, in the sense of Federer. We provide a representation formula for the Clarke generalized gradient using convex combinations and limits of gradients at differentiability points, which provides an extension of the well known Proximal Normal Formula. As an application, we prove for a suitable class of Hamilton-Jacobi equations that an a.e. solution whose epigraph has positive reach is indeed a viscosity solution. Joint work with Giovanni Colombo and Peter R. Wolenski.

We consider autonomous Tonelli Hamiltonians in two degrees of freedom. We prove that for many cohomology classes, the Mather set consists of periodic orbits, and the Aubry set equals the Mather set. Then we give some applications to C^0-integrable systems.

Spacetimes admitting a lightlike parallel null vector are in correspondence with Lagrangian mechanical systems on a quotient non-relativistic space. In this talk this correspondence is clarified by proving a formula which relates the Lorentzian distance on spacetime with the mechanical least action in the quotient non-relativistic space. Using the correspondence, the causal hierarchy for spacetimes can be read as a hierarchy of results on the mechanical least action. It is also shown that the Hamilton-Jacobi equation is related to the boundary of a special future set, and that the properties of the Lax-Oleinik semigroup can be deduced from results in Lorentzian geometry on the smoothness of null hypersurfaces. In the time independent case it is also possible to give a spacetime interpretation to the weak KAM theorem.

We will explain how we can construct C^1 functions whose set of critical points is homeomorphic to the unit interval [0,1]. We then present applications to the dynamic of Mañé Lagrangians.

We consider smooth deformations of compact Riemannian manifolds with or without boundary and the corresponding Laplace-Beltrami operators. We suppose that the initial metric is completely integrable or close to completely integrable. If the deformation is isospectral, we prove that that the average action on the KAM tori is constant along the deformation. As an application we prove infinitesimal rigidity of Liouville billiard tables. The proof is based on a construction of quasi-modes associated with KAM tori.

In this talk I will present geometric definitions for the Aubry and Mañé sets that allow us to understand in an intuitive way their property of symplectic invariance. With this purpose I will show that if H is an optical Hamiltonian an L is a Lipschitz exact Lagrangian manifold, isotopic to the zero section, contained in some sublevel of the Hamiltonian, say H less than or equal to some constant e, then there exists a Lipschitz exact Lagrangian graph contained in the same sublevel, such that the maximal invariant sets of L, on the e-level of H, and such graph coincide.

I will report on some joint work with R. de la Llave and E. Fontich in which we construct finite and infinite dimensional tori on lattices.

We consider a multivalued dynamics driven by the multivalued vector field F(x), with F convex compact valued in the n-dimensional Euclidean space and the dynamics asymptotically stable with respect to a compact attractor. We will focus on the case of a continuous F although the results hold for an upper semicontinuous multifunction. We associate to F the map Z(x) obtained for any x through the intersection of the polar cone of F (x) and the unit ball, and use it to define an in- trinsic lenght which indicates how any absolutely continuous curve is far from being a solution to the differential inclusion. By developing a geometric approach introduced by Fathi-Siconolfi we show that the attractor of the dynamics is the Aubry set associated to a certain quasi-convex Hamiltonian whose sublevel sets are given by Z. As a consequence, the classical converse Lyapunov theorem can be recovered, and a smooth Lyapunov function for the dynamics can be constructed. This talk is based on a joint research with Antonio Siconolfi.

We consider the homogenization of a periodic interfacial energy, such as in recents papers by Caffarelli and De La Llave, or Dirr, Lucia and Novaga. We provide a proof of a Gamma-limit, however, we also observe that thanks to the coarea formula, in most cases such a result is already known in the framework of BV homogenization. This leads to an interesting new construction for the plane-like minimizers in periodic media of Caffarelli and De La Llave, through a cell problem.

This is a joint work with F. Cagnetti and D. Gomes. We use the new Nonlinear Adjoint method recently introduced by Evans and a PDE approach to construct analogs to the Aubry-Mather measures for nonconvex Hamiltonians. These measures may fail to be invariant under the Hamiltonian flow and a dissipation, described by a positive semidefinite matrix of Borel measures, arises. However, in the important case of uniformly quasiconvex Hamiltonians, the dissipation vanishes, and as a consequence, the invariance is guaranteed.

We consider the zero angular momentum planar three body problem, and we fix the energy to a negative value. A syzygy is a three-body configuration where the three body are collinear. A result of R. Montgomery says that every solution with negative energy and zero angular momentum has a syzygy in forward time. For this reason, if we take a three body configuration on the zero velocity curve and the corresponding solution starting with zero velocity, at some time we will fond a syzygy. In this way, we define a map called "brake to syzygy" map. We will describe some interesting properties of it, and in particular, we construct an interesting brake periodic solution. Joint work with Rick Moeckel and Richard Montgomery.

Predicting turbulent flame speed is a fundamental problem in turbulent combustion theory. Several models have been proposed to study it, such as the G-equations, the F-equations (Majda-Souganidis model) and reaction-diffusion-advection (RDA) equations. In this talk, I will present applications of weak KAM theory in the study of the asymptotic growth of the ratio turbulent flame speed over turbulent intensity, when the turbulent intensity goes to infinity, for the above three models. This is part of a joint work with J. Xin.