Weak KAM beyond Hamilton-Jacobi – WKBHJ

This project is a continuation of the previous project ANR KAMFAIBLE.
Even if, in this project, there is some overlap with members of the previous project, there is an important renewal: of the 34 members of this project, only 11 were in the original previous project, and 10 among the new members are PhD students. This is due to the attractiveness of the subject, and to the fact that it is expanding its range of applicability in mathematics.
The goal of the previous project and this one is to bring together people from different horizons working in domains where the role of the Hamilton-Jacobi Equation is crucial: Lagrangian Dynamical Systems, PDE, Control, Celestial Mechanics, and Symplectic Geometry.
In the last 4 year during the previous project, a lot of activity in the subject took place, leading to more than 70 published papers by members of the previous project. There were several international conferences organized on that topic (and/or related topics) in various countries (US, Italy, Portugal, Japan...). We organized 2 big international conferences with a large number of people coming from abroad, Nice in February 2009, and Bordeaux in December 2011. There were also several smaller conferences also with participants from abroad.
With the financial resources provided in the last project emerged a cohesive body of mathematicians from various background and countries that are tightly working for further progress in the area, and to to extend the perimeter.
The work done under the previous project has generated new points of view, new problems and new directions in which to apply the tools developed. Therefore in this continuation project, besides deepening our previous achievements, we would like to emphasize the tools developed so far in our circle of ideas are now used in other domains like: Lyapunov functions of classical and multivalued dynamics, time functions, multidimensional Frenkel-Kontorova models, higher order PDE's, etc...
Some of the problems we want now to specifically address are: large time behavior of curvature type flows, large time behavior in degenerate second order equations, convergence of discounted solutions of the Hamilton-Jacobi Equation, higher regularity of invariant curves for twist maps, Mañé's conjecture, differentiability of beta functions, smoother critical subsolutions, weak KAM aspects of Frenkel-Kantorova models, infinite-dimensional weak KAM-theory, weak Kam theory for classical and multivalued (non-Lagrangian) dynamics (with applications to Lorentzian geometry), commuting non-smooth Hamiltonians, generalizations to non-convex Hamiltonians.
As our previous track record shows, we are well armed to deal with these problems.
An important part of these problems will be worked out with international collaborators (from US, Italy, England, Spain, Portugal, Mexico, Uruguay, Brazil...). We have also a lot of scientific contacts with people working in this area in China, Japan, Canada, Israel, Netherland, and Russia.
We will use the funds to organize several meetings (at least two of them large international meetings), and to work with our numerous foreign collaborators. We will also hire two Postdocs (in the last project we had two good postdocs: one is Maître de Conférences in Nice the other is Assistant professor in Penn State). We will also use funds for two PHD students (the group had 6 successful PhD students during the previous project). We will also disseminate our knowledge through summer schools.
Given the high international visibility of the previous project, we were able to raise money from national and international partners to enhance our activities (the US NSF, the Italian CNR, Research funds for visitors from Japan, Mexico, Brazil, Uruguay). If the ANR funding is adequate for the intensive international collaboration we have in mind, we plan to raise some matching funds for our international conferences and collaborations.