A dynamical approach of some Hamilton-Jacobi equations using weak KAM theory

Abstract:

Hamilton-Jacobi equations are first order PDEs of a special type related to the dynamics of a Hamiltonian flow. Weak KAM theory is a set of tools in dynamical systems that enable us to solve these equations using a semi-group of non-linear operators. We will develop in this lecture a discrete-in-time approach and compare this model to the Frenkel-Kontorova model in Solid State Physics. We will also show that the discrete model converges to the continuous model as the time step goes to zero.