A mini-lecture series consisting of four 1 hour lectures.
We would like to consider asymptotic behaviour of various problems set in cylinders.
Let be the simplest cylinder possible. A good model problem is the following. Consider the weak solution to
When is it trues that the solution converges toward
the solution of the lower dimensional problem below ?
If so in what sense ? With what speed of convergence with respect to ? What happens when is also allowed to depend on ? What happens if is periodic in , 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.