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

In $L_2({\mathbb{R}}^3;{\mathbb{C}}^3)$, we consider a selfadjoint operator ${\mathcal{L}}_{\epsilon }$, $\epsilon >0$, 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 $\cos (\tau {\mathcal{L}}_{\epsilon }^{1\mathrm{/}2})$ and ${\mathcal{L}}_{\epsilon }^{-1\mathrm{/}2}\sin (\tau {\mathcal{L}}_{\epsilon }^{1\mathrm{/}2})$ for $\tau \in \mathbb{R}$ and small $\epsilon$. 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 $\eta (\mathbf{x}\mathrm{/}\epsilon )$.