Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain with boundary of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, consider a matrix elliptic second-order differential operator $A_{D,\varepsilon}$ with Dirichlet boundary condition. Here $\varepsilon >\nobreak0$ is a small parameter; the coefficients of $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The operator $A_{D,\varepsilon}^{-1}$ in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ is approximated with an error of order $\varepsilon^{1/2}$. The approximation is given by the sum of the operator $(A^0_D)^{-1}$ and a first-order corrector. Here $A^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.