Precised approximations in elliptic homogenization beyond the periodic setting

Abstract

We consider homogenization problems for linear elliptic equations in divergence form. The coefficients are assumed to be a local perturbation of some periodic background. We prove W 1,p and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and defined in [10,11], and apply the same strategy of proof as Avellaneda and Lin in [1]. We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.

Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

“…In (4), the corrector employed is the function w p solution to the corrector equation (7), the existence and uniqueness (up to additive constants) of which has been established in our previous works [3,4]. Such a corrector is different from the periodic corrector, and although intuitively one could have thought, based on the observation that the presence of a does not modify the homogenized equation (2), that the periodic corrector w per p would give an equally accurate approximation, it does not. The rates of convergence obtained are made precise in our main result, namely Théorème 2.1 of the French version, and estimates (10) through (12) in various norms.…”

“…In passing, and as is also the case in the proof of the periodic case, we establish estimates for the Green function G ǫ of the original problem (see (32), (33), (34) of the French version) and its convergence to the (possibily corrected) Green function of the homogenized problem (2) (see (28)). The details of the results announced in this Note are given in our publications [2,12].…”

“…L'objet de cette Note est d'énoncer précisément ce résultat, et de donner les grandes lignes de sa preuve : « le correcteur corrige » dans la norme considérée, à un ordre précisé. Les résultats annoncés dans cette Note, et leurs preuves, seront détaillés dans les publications [2,12] en préparation.…”

“…Une extension à un cas abstrait "général" est mentionnée. Les résultats annoncés dans cette Note seront précisés dans les documents [2,12].…”

“…Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [13], and borrows a lot from it. The details of the results announced in this Note are given in our publications [2,12].…”