Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class H~1(R~d)

摘要

Investigation of a class of matrix periodic elliptic second-order differential operators A_ε in R~d with rapidly oscillating coefficients (depending on X/ε) is continued. The homogenization problem in the small period limit is studied. Approximation for the resolvent (A_ε+I)~(-1) in the operator norm from L_2(R~d) to H~1(R~d) is obtained with an error of order ε. In this approximation, a corrector is taken into account. Moreover, the (L_2→L_2)-approximations of the so-called fluxes are obtained.