A field theory for symplectic fibrations over surfaces.
A finite dimensional linear programming approximation of Mather's variational problem
We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.
Action minimizing invariant measures for positive definite Lagrangian systems.
Adjoint methods for obstacle problems and weakly coupled systems of PDE
The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton − Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.
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 proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.
Décomposition des difféomorphismes du tore en applications déviant la verticale (avec un appendice en collaboration avec Jean-Marc Gambaudo)
Dénombrement de géodésiques fermées, sous contraintes homologiques
Effective Hamiltonians and Quantum States
We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function solving the eikonal equation aėȧnd a probability measure solving a related transport equation.We present some elementary formal identities relating certain quantum states and . We show also how to build out of an approximate...
Generic uniqueness of minimal configurations with rational rotation numbers in Aubry-Mather theory.
Invariance of global solutions of the Hamilton-Jacobi equation
We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.
Linear programming interpretations of Mather’s variational principle
We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an -dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].
Linear programming interpretations of Mather's variational principle
We discuss some implications of linear programming for Mather theory [13-15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an n-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [5-8].
Multichain-type solutions for Hamiltonian systems.
Nouvelle preuve d’un théorème de Yuan et Hunt
Un théorème de Guo-Cheng Yuan Brian R. Hunt affirme que, pour mesure de probabilité invariante d’un système dynamique hyperbolique , les fonctions lipschitziennes pour lesquelles est minimisante ont un intérieur non vide (en topologie de Lipschitz) si et seulement si est une orbite périodique de . Je donnerai une nouvelle preuve de ce théorème, ou plutôt d’un énoncé essentiellement équivalent. Je discuterai aussi de la stabilité des orbites périodiques minimisantes de grande période.
On the mobility and efficiency of mechanical systems
It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.
On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics
We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...
Optimalité systolique infinitésimale de l’oscillateur harmonique
Nous étudions les aspects infinitésimaux du problème suivant. Soit un hamiltonien de dont la surface d’énergie borde un domaine compact et étoilé de volume identique à celui de la boule unité de . La surface d’énergie contient-elle une orbite périodique du système hamiltoniendont l’action soit au plus ?
Perturbation theory and discrete Hamiltonian dynamics.
Regularity of solutions of the Hamilton-Jacobi equation
Stable norms of non-orientable surfaces
We study the stable norm on the first homology of a closed non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.