Introduction to Weak KAM theory and Hamilton-Jacobi equations

Many dynamical systems are governed by Hamiltonian systems of ODEs. For some very simple systems such as two bodies interacting by gravitational forces or such as harmonic oscillators, it is possible to explicitly integrate the equations of motion. In this cases it is possible to find a system of coordinates (ρ, Θ) called the action-angle coordinates in which the dynamics reduces to ρ^. = 0 and Θ^. = constant. It is then natural to ask if for systems which are small perturbations of integrable systems, such as the three body problem, the same kind of simple dynamics persists. The classical Kolmogorov-Arnold-Moser (KAM) Theorem, asserts that indeed, for such small perturbations most of these invariant tori ρ = constant will survive and that moreover, on these tori, the dynamic is still very simple. Weak KAM theory, which was pioneered by Fathi and Mather, tries to understand what remains of this picture when the perturbations are not small anymore. This theory is intimately connected with the theory of homogenization for Hamilton-Jacobi equations developed by Lions-Papanicolaou-Varadhan. After giving a short introduction to the classical KAM theory, we will study the notion of viscosity solutions for Hamilton-Jacobi equations and their homogenization. Among the topics that will be covered, there will be:

• Aubry-Mather theory,
• Connection with optimal transport (following Bernard-Buoni),
• Aubry-Mather theory for minimal surfaces,
• Stochastic Weak KAM theory and applications to homogenization.

Date and time info
Thursday 13.30-15.00

Keywords
Dynamical systems, KAM theory, Hamilton-Jacobi equations, viscosity solutions, homogenization, Aubry-Mather theory

Prerequisites
You should know about measure theory, basic ODEs