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

Abstract:

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.