Resolvent approximations in $L^2$-norm for elliptic operators acting in a perforated space

We study homogenization of a second-order elliptic differential operator $A_\varepsilon=-\mathrm{div}\, a(x/\varepsilon)\nabla$ acting in an $\varepsilon$-periodically perforated space, where $\varepsilon$ is a small parameter. Coefficients of the operator $A_\varepsilon$ are measurable $\varepsilon$-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent $(A_\varepsilon+1)^{-1}$ with remainder term of order $\varepsilon^2$ as $\varepsilon\to 0$ in operator $L^2$-norm on the perforated space. This approximation turns to be the sum of the resolvent $(A_0+1)^{-1}$ of the homogenized operator $A_0=-\mathrm{div}\, a^0\nabla,$ $a^0>0$ being a constant matrix, and some correcting operator $\varepsilon \mathcal{C}_\varepsilon.$ The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.