Homogenization of the Stationary Maxwell System with Periodic Coefficients in a Bounded Domain

Abstract

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 $\eta({\mathbf x}/ \varepsilon )$ 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 $\varepsilon >0$ is the small parameter. We use the following notation for the solutions of the Maxwell system: ${\mathbf u}_\varepsilon$ is the electric field intensity, ${\mathbf v}_\varepsilon$ is the magnetic field intensity, ${\mathbf w}_\varepsilon$ is the electric displacement vector, and ${\mathbf z}_\varepsilon$ is the magnetic displacement vector. It is known that ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ weakly converge in $L_2({\mathcal O})$ 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 $\varepsilon \to 0$. We improve the classical results and find approximations for ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ in the $L_2({\mathcal O})$-norm. The error terms do not exceed $C \sqrt{\varepsilon} (\| {\mathbf q}\|_{L_2}+\|{\mathbf r}\|_{L_2})$, where the divergence free vector-valued functions ${\mathbf q}$ and ${\mathbf r}$ are the right-hand sides of the Maxwell equations.