Homogenization with corrector term for periodic elliptic differential operators

We continue to study the class of matrix periodic elliptic differential operators Aε in Rd with coefficients oscillating rapidly (i.e., depending on x/ε). This class was introduced in the authors’ earlier work of 2001 and 2003. The problem of homogenization in the small period limit is considered. Approximation for the resolvent (Aε + I)−1 is obtained in the operator norm in L2(R) with error term of order ε2. The so-called corrector is taken into account. We develop the approach of our paper of 2003, where approximation with no corrector term but with remainder term of order ε was found. The paper is based on the operator-theoretic material obtained in our paper in the previous issue of this journal. Though the present paper is a continuation of the earlier work, it can be read independently.