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 $({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1}$ converges in the $L_2({\mathcal O};{\mathbb C}^n)$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order error 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.