Séminaire d'AnalyseThe Aubry-Mather theory and difference Hamilton-Jacobi equations
jeudi 24 juin 2010 11:00 - Tours - Salle 2290 (Bât E2)
We consider periodically time-dependent Hamiltonian systems with one degree of freedom, the corresponding Hamilton-Jacobi equations and scalar conservation laws. The relation among them is made clear by Albert Fathi and Weinan E. In this talk we focus our attention on numerical aspects of the issue. First we see results of difference approximation of periodic entropy solutions to the conservation laws and apply them to the computation of the Aubry-Mather sets. Then we translate the approximation into that of viscosity solutions of the Hamilton-Jacobi equations and find Aubry-Mather theory like relation among these approximate objects. The key tool of our arguments is a stochastic and variational representation of difference solutions which corresponds to the variational representation of viscosity solutions by the value function.