2020/08/07 by Dorodnyi, Mark, Suslina, Tatiana
#35B27 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2008.03047
In L2(\mathbb R3;\mathbb C3), we consider a selfadjoint operator \mathcal Lε, ε >0, given by the differential expression μ0-1/2curl η(x/ε)-1 curl μ0-1/2 - μ01/2∇ ν(x/ε) div μ01/2, where μ0 is a constant positive matrix, a matrix-valued function η(x) and a real-valued function ν(x) are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions cos (τ\mathcal Lε1/2) and \mathcal Lε-1/2 sin (τ\mathcal Lε1/2) for τ∈ \mathbb R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator \mathcal L0 in the norm of operators acting from the Sobolev space Hs (with a suitable s) to L2. Here \mathcal L0 is the effective operator with constant coefficients. Also, an approximation with corrector in the (Hs → H1)-norm for the operator \mathcal Lε-1/2 sin (τ\mathcal Lε1/2) is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on τ. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to μ0, and the dielectric permittivity is given by the matrix η(x/ε).