Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain

In a bounded domain $\mathcal{O}\subset {\mathbb{R}}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by $\eta (\mathbf{x}\mathrm{/}\epsilon )$ and $\mu({\mathbf x}/ \varepsilon )$,where$\eta( {\mathbf x})$and$\mu({\mathbf x})$ are symmetric $(3\times 3)$-matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here $\epsilon >0$ is the small parameter. We use the following notation for the solutions of the Maxwell system: ${\mathbf{u}}_{\epsilon }$ is the electric field intensity, ${\mathbf{v}}_{\epsilon }$ is the magnetic field intensity, ${\mathbf w}\varepsilon$istheelectricdisplacementvector,and${\mathbf z}\varepsilon$isthemagneticdisplacementvector\mathrm{.}Itisknownthat${\mathbf u}\varepsilon$,${\mathbf v}\varepsilon$,${\mathbf w}\varepsilon$,and${\mathbf z}\varepsilon$weaklyconvergein$L_2({\mathcal O})$ to the corresponding homogenized fields ${\mathbf{u}}_0$, ${\mathbf{v}}_0$, ${\mathbf w}0$,and${\mathbf z}0$ (the solutions of the homogenized Maxwell system with the effective coefficients), as $\epsilon \to 0$. We improve the classical results and find approximations for ${\mathbf{u}}_{\epsilon }$, ${\mathbf{v}}_{\epsilon }$, ${\mathbf{w}}_{\epsilon }$, and ${\mathbf z}\varepsilon$inthe$L_2({\mathcal O})$-norm\mathrm{.}Theerrortermsdonotexceed$C \sqrt{\varepsilon} (| {\mathbf q}|{L_2}+|{\mathbf r}|_{L_2})$, where the divergence free vector-valued functions $\mathbf{q}$ and $\mathbf{r}$ are the right-hand sides of the Maxwell equations.