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 $\eta({\mathbf x}/ \varepsilon)$, 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 $\varepsilon >0$ 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 $\varepsilon \to 0$, the solutions of the Maxwell system, namely, the electric field intensity ${\mathbf u}_\varepsilon$, the electric displacement vector ${\mathbf w}_\varepsilon$, the magnetic field intensity ${\mathbf v}_\varepsilon$, and the magnetic displacement vector ${\mathbf z}_\varepsilon$ 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 ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ converge to ${\mathbf v}_0$ and ${\mathbf z}_0$, respectively, in the $L_2$-norm, the error terms do not exceed $C \varepsilon \| {\mathbf r}\|_{L_2}$. We also find approximations for ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ in the energy norm with error $C\sqrt{\varepsilon} \|{\mathbf r}\|_{L_2}$. For ${\mathbf u}_\varepsilon$ and ${\mathbf w}_\varepsilon$ we obtain approximations in the $L_2$-norm with error $C\sqrt{\varepsilon} \| {\mathbf r}\|_{L_2}$.