Operator Estimates in Homogenization of Elliptic Systems of Equations

Abstract

We study homogenization of nonselfadjoint second order elliptic systems with ε-periodic rapidly oscillating coefficients as ε → 0. We obtain the L²- and H¹-estimates for the homogenization error of order ε. The estimates admit the operator form and can be written in terms of the resolvents of the original and approximate systems in the operator norm \( {\left\Vert \cdot \right\Vert}_{L^2\to {L}^2} \) or \( {\left\Vert \cdot \right\Vert}_{L^2\to {H}^1} \). The shift method is used for obtaining such estimates.