Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem

In a bounded domain ${\mathcal O} \subset {\mathbb R}^d$ with $C^{1,1}$ boundary a matrix elliptic second-order operator ${A}_{D,\varepsilon}$ with Dirichlet boundary condition is studied. The coefficients of this operator are periodic and depend on $\mathbf{x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. The sharp-order error estimate $\|{A}_{D,\varepsilon}^{-1} - ({A}_D^0)^{-1} \|_{L_2 \to L_2} \le C \varepsilon$ is obtained. Here ${A}^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.