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 ${\mathcal{L}}_{\epsilon }$, $\epsilon >0$, given by the differential expression ${\mu }_0^{-1/2}\mathrm{curl}\eta (\mathbf{x}/\epsilon {)}^{-1}\mathrm{curl}{\mu }_0^{-1/2}-{\mu }_0^{1/2}\nabla \nu (\mathbf{x}/\epsilon )\mathrm{div}{\mu }_0^{1/2}$, 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 $\cos (\tau {\mathcal{L}}_{\epsilon }^{1/2})$ and ${\mathcal{L}}_{\epsilon }^{-1/2}\sin (\tau {\mathcal{L}}_{\epsilon }^{1/2})$ 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 ${\mathcal{L}}_{\epsilon }^{-1/2}\sin (\tau {\mathcal{L}}_{\epsilon }^{1/2})$ 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 $\eta (\mathbf{x}/\epsilon )$.

29 pages