Operator error estimates for homogenization of the elliptic dirichlet problem in a bounded domain

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