Local profiles for elliptic problems at different scales

This is joint work with Xavier Blanc (University Denis Diderot, Paris) and Pierre-Louis Lions (College de France, Paris). We present a general approach to approximate at the fine scale the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a ”nice” (in the simplest possible case say periodic) function which is, in some sense to be made precise, perturbed. The approach is based on the determination of a local profile, solution to an equation similar to the corrector equation in classical homogenization. We prove that this equation has a unique solution, in various functional settings depending upon the perturbation: local perturbation, two different periodic structures separated by a common interface, etc.