Homogenization of a nonstationary model equation of electrodynamics

In L2(ℝ3;ℂ3), we consider a self-adjoint operator ℒε, ε > 0, generated by the differential expression curl η(x/ε)−1 curl−∇ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τℒε1/2) and ℒε−1/2 sin(τℒε1/2) for τ ∈ ℝ and small ε. It is shown that these operators converge to cos(τ(ℒ0)1/2) and (ℒ0)−1/2 sin(τ(ℒ0)1/2), respectively, in the norm of the operators acting from the Sobolev space Hs (with a suitable s) to ℒ2. Here ℒ0 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 ∂τ2vε = −ℒεvε, div vε = 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 η(x/ε).