Homogenization of nonstationary periodic Maxwell system in the case of constant permeability

Abstract

In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $\mu_0^{-1/2}\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} \mu_0^{-1/2} - \mu_0^{1/2}\nabla \nu(\mathbf{x}/\varepsilon) \operatorname{div} \mu_0^{1/2}$, where $\mu_0$ is a constant positive matrix, a matrix-valued function $\eta(\mathbf{x})$ and a real-valued function $\nu(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (\tau {\mathcal L}_\varepsilon^{1/2})$ and ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ for $\tau \in {\mathbb R}$ and small $\varepsilon$. It is shown that these operators converge to the corresponding operator-valued functions of the operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the effective operator with constant coefficients. Also, an approximation with corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on $\tau$. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to $\mu_0$, and the dielectric permittivity is given by the matrix $\eta(\mathbf{x}/\varepsilon)$.