Stratified Problems arising from Homogenization of Hamilton-Jacobi equations

par Yves Achdou

lundi 16 déc. 2024, 11:00 → 12:00 Europe/Paris

3L15 (Laboratoire de Mathématiques d'Orsay)

We describe several cases in which the homogenization of stationary Hamilton-Jacobi equations  leads to stratified problems (the class of stratified problems has been introduced by A. Bressan and Y. Hong and later studied by G. Barles and E. Chasseigne). We first recall results obtained with S. Oudet and N. Tchou, in which the limiting problem in $\mathbb R^d$ is a stratified problemin which  the stratification is of the type $(\mathbb R^d\setminus\mathcal M_{d-1}) \cup \mathcal M_{d-1}$, where $\mathcal M_{d-1}$ is a $d-1$ dimensional subspace. Next, we  consider a class of  Hamilton-Jacobi equations in  which the  Hamiltonian is obtained by perturbing  near the origin an otherwise periodic Hamiltonian (collaboration with C. Le Bris).  The limiting problem involves the stratification $(\mathbb R^d\setminus\{0\}) \cup \{0\}$.

Finally, we study homogenization  for a class of bidimensional stationary Hamilton-Jacobi equations where the  Hamiltonian is obtained by perturbing near a half-line  a Hamiltonian which does not depend on the fast variable, or depends on the fast variable in a periodic manner. We prove that the limiting problem involves a  stratification, made of a submanifold of dimension zero, namely the endpoint of the half-line,  a submanifold of dimension one, the open half-line, and the complement of the latter two sets which is a submanifold of dimension two. The limiting problem involves effective Hamiltonians which are associated to the  abovementioned three submanifolds and keep track of the perturbation.