Homogenization of a stationary periodic Maxwell system in a bounded domain in the case of constant magnetic permeability

Abstract

In a bounded domain $\mathcal{O}\subset {\mathbb{R}}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the boundary conditions of perfect conductivity. It is assumed that the magnetic permeability is given by a constant positive $(3\times 3)$-matrix ${\mu }_0$ and the dielectric permittivity is of the form $\eta (\mathbf{x}/\epsilon )$, where $\eta (\mathbf{x})$ is a $(3\times 3)$-matrix-valued function with real entries, periodic with respect to some lattice, bounded and positive definite. Here $\epsilon >0$ is the small parameter. Suppose that the equation involving the curl of the magnetic field intensity is homogeneous, and the right-hand side $\mathbf{r}$ of the second equation is a divergence-free vector-valued function of class $L_2$. It is known that, as $\epsilon \to 0$, the solutions of the Maxwell system, namely, the electric field intensity ${\mathbf{u}}_{\epsilon }$, the electric displacement vector ${\mathbf{w}}_{\epsilon }$, the magnetic field intensity ${\mathbf{v}}_{\epsilon }$, and the magnetic displacement vector ${\mathbf{z}}_{\epsilon }$ weakly converge in $L_2$ to the corresponding homogenized fields ${\mathbf{u}}_0$, ${\mathbf{w}}_0$, ${\mathbf{v}}_0$, ${\mathbf{z}}_0$ (the solutions of the homogenized Maxwell system with effective coefficients). We improve the classical results. It is shown that ${\mathbf{v}}_{\epsilon }$ and ${\mathbf{z}}_{\epsilon }$ converge to ${\mathbf{v}}_0$ and ${\mathbf{z}}_0$, respectively, in the $L_2$-norm, the error terms do not exceed $C\epsilon \Vert \mathbf{r}{\Vert }_{L_2}$. We also find approximations for ${\mathbf{v}}_{\epsilon }$ and ${\mathbf{z}}_{\epsilon }$ in the energy norm with error $C\sqrt{\epsilon }\Vert \mathbf{r}{\Vert }_{L_2}$. For ${\mathbf{u}}_{\epsilon }$ and ${\mathbf{w}}_{\epsilon }$ we obtain approximations in the $L_2$-norm with error $C\sqrt{\epsilon }\Vert \mathbf{r}{\Vert }_{L_2}$.

28 pages