Marianne Olsson

Very weak multiscale convergence

We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.

Some Remarks on Homogenization in Perforated Domains

Mathematical homogenization theory deals with the question of finding effective properties and microvariations in heterogeneous materials. Usually, the difficulty consists in handling the rapid periodic oscillations of coefficients governing some partial differential equation. Sometimes, though, there are also many small periodically arranged holes in the material, i.e., the domain of the equation. In this latter case we have to distinguish between the situations where the holes have Neumann (e.g., isolating holes) and Dirichlet (e.g., constant temperature) boundary conditions. The aim of this chapter is to investigate an intermediate case, where holes with constant zero temperature are coated with a thin layer of a material with low heat-conduction number.