Homogenization of a stationary periodic Maxwell system in a bounded domain in the case of constant magnetic permeability
Tatiana Aleksandrovna Suslina · St Petersburg Mathematical Journal · 2019
In a bounded domain $\mathcal {O}\subset \mathbb {R}^3$ of class $C^{1,1}$, 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 $\mathbf {z}_\varepsilon$ converge weakly 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). Classical results are improved. 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, and the error terms do not exceed $C \varepsilon \|\mathbf {r}\|_{L_2}$. Approximations for $\mathbf {v}_\varepsilon$ and $\mathbf {z}_\varepsilon$ in the energy norm are also found with error $C\sqrt {\varepsilon } \|\mathbf {r}\|_{L_2}$. For $\mathbf {u}_\varepsilon$ and $\mathbf {w}_\varepsilon$, approximations in the $L_2$-norm with error $C\sqrt {\varepsilon } \|\mathbf {r}\|_{L_2}$ are also found.