Homogenization of a nonstationary model equation of electrodynamics

Abstract

In L₂(ℝ₃;ℂ₃), we consider a self-adjoint operator ℒ_ε, 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(τℒ_ε^1/2) and ℒ_ε^−1/2 sin(τℒ_ε^1/2) 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 H^s (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 ∂_τ²v_ε = −ℒ_εv_ε, div v_ε = 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).