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.
Categories: math.AP