Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 3 of 4

Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 3 of 4

Fri, 24/05
10:00 		Michel Chipot (University of Zurich) OxPDE Special Seminar Add to calendar Gibson Grd floor SR
Asymptotic Behavior of Problems in Cylindrical Domains - Lecture 3 of 4
A mini-lecture series consisting of four 1 hour lectures. We would like to consider asymptotic behaviour of various problems set in cylinders. Let $ \Omega_\ell = (-\ell,\ell)\times (-1,1) $ be the simplest cylinder possible. A good model problem is the following. Consider $ u_\ell $ the weak solution to
$$ \cases{ -\partial_{x_1}^2 u_\ell - \partial_{x_2}^2 u_\ell = f(x_2) \quad \hbox{in } \Omega_\ell, \quad \cr \cr u_\ell = 0 \quad \hbox{ on } \quad \partial \Omega_\ell. \cr} $$
When $ \ell \to \infty $ is it trues that the solution converges toward $ u_\infty $ the solution of the lower dimensional problem below ?
$$ \cases{ - \partial_{x_2}^2 u_\infty = f(x_2) \quad \hbox{in }(-1,1), \quad \cr \cr u_\infty = 0 \quad \hbox{ on } \quad \partial (-1,1). \cr} $$
If so in what sense ? With what speed of convergence with respect to $ \ell $ ? What happens when $ f $ is also allowed to depend on $ x_1 $ ? What happens if $ f $ is periodic in $ x_1 $, is the solution forced to be periodic at the limit ? What happens for general elliptic operators ? For more general cylinders ? For nonlinear problems ? For variational inequalities ? For systems like the Stokes problem or the system of elasticity ? For general problems ? ... We will try to give an update on all these issues and bridge these questions with anisotropic singular perturbations problems. Prerequisites : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.