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 (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.
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