Hamilton-Jacobi et théorie KAM faible : à l'interface des EDP, systèmes dynamiques lagrangiens et symboliques – KAMFAIBLE

The goal of our project is to bring together the French mathematicians who work in areas related to the Hamilton-Jacobi Equation: Dynamical Systems, PDE, and Optimal Control in order to enhance further progress in this subject which has known dramatic development in the last ten years. The area now is known by the name «Weak KAM Theory». It should be clear that one of the originalities of the project is that people from quite different horizon in mathematics are already collaborating together on problems which necessite to be solved far apart mathematical tools. Moreover, the circle of ideas in weak KAM theory is finding more and more domains where it can be applied. Originally Hamilton and Jacobi set up this Hamilton-Jacobi Equation to be able to find orbits in Classical Mechanical Systems, which are now part of the theory of Lagrangian Dynamical Systems. Basically for a long time this PDE was used in Dynamical Systems as a local technique, via the Method of Characteristics. This was due to the absence of global smooth (at least C1) solutions. Of course, interest in the PDE aspects was also immense especially when it was realized that a lot of problems in Optimal Control could be formulated in that framework. In the 1980's both the theory of Lagrangian Dynamical Systems, and the PDE theory of the Hamilton-Jacobi Equation had known dramatic progress on the global aspects, due to the discovery of Aubry Mather sets, on the dynamical side, and to the introduction of viscosity solutions on the PDE side. By the end of 1990's the connection between Aubry-Mather sets and viscosity solutions was discovered. It generated a lot of work: asymptotic behavior of the Lax-Oleinik semi-group, existence of homoclinic orbits, stationary ergodic homogenization, relation with the Aronsson-Euler Equation, progress in the study of solutions obtained by the vanishing viscosity method. Weak KAM theory has since then seen its scope extended into different areas: Ergodic Optimization, discret viscosity solutions, relation with Monge Transportation Problem, Lyapunov functions, time-functions on Lorentz manifolds. This progress was only made possible by interactions between different mathematical subjects. There are still many attractive and interdisciplinary open problems in the area. We propose to work on some of them, not only the team members, but also Post-Docs, that we hope to hire on this contract, and PhD students, for whom there are nice and workable problems. Some of the problems we want to address are: regularity of solutions of Hamilton-Jacobi equation, asymptotic behavior of Lax-Oleinik semi-groups for unbounded solutions, Ergodic Optimization, non-convex Hamiltonians, generalizing the theory to domains with non-empty boundary, applications to Celestial Mechanics, and to Control problems. We will use the funds to meet regularly, invite mathematicians for abroad, organize 2 International conferences, and disseminate our knowledge through advanced courses given worldwide.