On a selection principle in the ergodic approximation of the Hamilton-Jacobi equation
Homogenization and Random Phenomenon
21 October 14:00 - 14:55
Andrea Davini - University of Rome
The so called ergodic approximation is a technique introduced by Lions, Papanicolaou and Varadhan in 1987 to show the existence of solutions to the cell problem, i.e. a nonlinear PDE of the kind H(x,Du)=c set on the torus, where c is a constant uniquely associated with the Hamiltonian H. The idea is that of adding a term consisting of u multiplied by a positive parameter and of studying the asymptotic behavior of the solution of the corresponding equation as the parameter goes to zero. Under standard coercivity assumptions on the Hamiltonian, such solutions converge, along subsequences, to a solution of the cell problem, which a priori depends on the chosen subsequence. When the Hamiltonian is convex in the momentum, we show that a unique solution is selected at the limit and we characterize it in terms of a class of probability measures introduced in the framework of weak KAM Theory. This is a joint work with A. Fathi, R. Iturriaga and M. Zavidovique.
KTH Royal Institute of Technology
The University of Chicago