Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients

Abstract

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter; the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1}$, $\lambda \in {\mathbb C} {\setminus} {\mathbb R}_+$, converges in the $L_2$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order estimate $\|({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1} - ({\mathcal A}_{N}^0 -\lambda I)^{-1}\|_{L_2\to L_2} \le C\varepsilon $ is obtained. Approximation for the resolvent $({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1}$ 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 $O(\sqrt{\varepsilon})$ is found. Approximation is given by the sum of the operator $({\mathcal A}^0_N - \lambda I)^{-1}$ and the first order corrector. In a strictly interior subdomain ${\mathcal O}'$ a similar approximation with an error $O(\varepsilon)$ is obtained. The case where $\lambda=0$ is also studied.