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. \smallskip \noindent {\bf Prerequisites} : Elementary knowledge on Sobolev Spaces and weak formulation of elliptic problems.