Precised approximations in elliptic homogenization beyond the periodic setting
Xavier Blanc, Marc Josien, Claude Le Bris · Asymptotic Analysis · 2019
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 [Formula: see text] 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.