We introduce a notion of two-scale convergence in the space \(L^{2}(\Omega)\) of square integrable functions (more generally we can consider \(L^{p}(\Omega)\)). When we need to treat different scales in problems of applications, the strong and weak convergences are inadequate. There are different applications including problems from homogenization where we see the appearance of multi-scales. We also state a compactness theorem in two-scale (or multi-scale) convergence which was introduced in the eighties.
A more general approach is the method of unfolding. Again unfolding operators have been introduced and used to study homogenization problems which were more general than two scale convergence. It was introduced for problems with rapidly oscillating coefficients and porous domains. Later, this was developed for domains with various types of oscillating boundaries. Our research group has developed many new unfolding operators and it has been applied to several homogenization problems of interest. We briefly discuss some of the developments from our group.
