Precised approximations in elliptic homogenization beyond the periodic setting

Article type: Research Article

Authors: Blanc, X.^{a; *} | Josien, M.^b | Le Bris, C.^b

Affiliations: [a] Université Paris-Diderot, Sorbonne Paris-Cité, Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, F-75013 Paris, France. E-mail: blanc@ann.jussieu.fr | [b] Ecole des Ponts and INRIA, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France. E-mails: claude.le-bris@enpc.fr, marc.josien@enpc.fr

Correspondence: [*] Corresponding author. E-mail: blanc@ann.jussieu.fr.

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 W1,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 (Comm. Partial Differential Equations 40 (2015) 2173–2236; Comm. Partial Differential Equations 43 (2018) 965–997), and apply the same strategy of proof as Avellaneda and Lin in (Comm. Pure Appl. Math. 40 (1987) 803–847). We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.

Keywords: Homogenization, elliptic PDE, periodic media, defects, convergence rate

DOI: 10.3233/ASY-191537

Journal: Asymptotic Analysis, vol. 116, no. 2, pp. 93-137, 2020