Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures

Xavier Blanc (1) , C Le Bris (2) , Pierre Louis Lions (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 MATHERIALS - MATHematics for MatERIALS
3 Collège de France - Chaire Équations aux dérivées partielles et applications

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].

Mots clés: Homogenization; interface; defects; quasiperiodic; elliptic PDE

Domaines: Equations aux dérivées partielles [math.AP]

Xavier Blanc, C Le Bris, Pierre Louis Lions. Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures. Communications in Partial Differential Equations, 2015, ⟨10.1080/03605302.2015.1043464⟩. ⟨hal-01143193v2⟩