Following-up on our previous work [10], 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. The well-posedness of that equation, in various functional settings depending upon the nature of the perturbation, is the purpose of this article. The case of a local perturbation is first addressed. The case of a more complex geometrical structure (such as the prototypical case of two different periodic structures separated by a common interface) is next discussed. Some related problems, and future directions of research are mentioned. Most of the results presented here have been announced in [28] and summarized in [11].