Homogenization of a Nonstationary Model Equation of Electrodynamics

In $L_2(\mathbb R^3;\mathbb C^3)$, we consider a self-adjoint operator $\mathscr L_\varepsilon$, $\varepsilon >0$, generated by the differential expression $\operatorname{curl}\eta(\mathbf x /\varepsilon)^{-1}\operatorname{curl} -\nabla\nu(\mathbf x/\varepsilon)\operatorname{div}$. Here the matrix function $\eta(\mathbf x)$ with real entries and the real function $\nu(\mathbf x)$ are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators $\cos(\tau\mathscr L_\varepsilon^{1/2})$ and $\mathscr L_\varepsilon^{-1/2} \sin(\tau\mathscr L_\varepsilon^{1/2})$ for $\tau\in\mathbb R$ and small $\varepsilon$. It is shown that these operators converge to $\cos(\tau(\mathscr L^0)^{1/2})$ and $(\mathscr L^0)^{-1/2}\sin(\tau(\mathscr L^0)^{1/2})$, respectively, in the norm of the operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here $\mathscr L^0$ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation $\partial^2_\tau\mathbf v_\varepsilon =-\mathscr L_\varepsilon\mathbf v_\varepsilon$, $\operatorname{div}\mathbf v_\varepsilon=0$, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to $1$ and the dielectric permittivity is given by the matrix $\eta(\mathbf x/\varepsilon)$.