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

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