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 $\eta (\mathbf{x}/\epsilon )$ and $\mu (\mathbf{x}/\epsilon )$, 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, ${\mathbf{w}}_{\epsilon }$ is the electric displacement vector, and ${\mathbf{z}}_{\epsilon }$ is the magnetic displacement vector. It is known that ${\mathbf{u}}_{\epsilon }$, ${\mathbf{v}}_{\epsilon }$, ${\mathbf{w}}_{\epsilon }$, and ${\mathbf{z}}_{\epsilon }$ 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 $\epsilon \to 0$. We improve the classical results and find approximations for ${\mathbf{u}}_{\epsilon }$, ${\mathbf{v}}_{\epsilon }$, ${\mathbf{w}}_{\epsilon }$, and ${\mathbf{z}}_{\epsilon }$ in the $L_2(\mathcal{O})$-norm. The error terms do not exceed $C\sqrt{\epsilon }(\Vert \mathbf{q}{\Vert }_{L_2}+\Vert \mathbf{r}{\Vert }_{L_2})$, where the divergence free vector-valued functions $\mathbf{q}$ and $\mathbf{r}$ are the right-hand sides of the Maxwell equations.

42 pages. arXiv admin note: text overlap with arXiv:1810.11328