TY - JOUR

T1 - Homogenization of a non-stationary periodic Maxwell system in the case of constant permeability

AU - Суслина, Татьяна Александровна

AU - Дородный, Марк Александрович

N1 - Publisher Copyright: © 2021 Elsevier Inc.

PY - 2022/1/15

Y1 - 2022/1/15

N2 - In L 2(R 3;C 3), we consider a selfadjoint operator L ε, ε>0, given by the differential expression μ −1/2curlη(x/ε) −1curlμ −1/2−μ 1/2∇ν(x/ε)divμ 1/2, where μ 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⁡(τL ε 1/2) and L ε −1/2sin⁡(τL ε 1/2) for τ∈R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator L 0 in the norm of operators acting from the Sobolev space H s (with a suitable s) to L 2. Here L 0 is the effective operator with constant coefficients. Also, an approximation with corrector in the (H s→H 1)-norm for the operator L ε −1/2sin⁡(τL ε 1/2) 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 non-stationary Maxwell system in the case where the magnetic permeability is equal to μ, and the dielectric permittivity is given by the matrix η(x/ε).

AB - In L 2(R 3;C 3), we consider a selfadjoint operator L ε, ε>0, given by the differential expression μ −1/2curlη(x/ε) −1curlμ −1/2−μ 1/2∇ν(x/ε)divμ 1/2, where μ 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⁡(τL ε 1/2) and L ε −1/2sin⁡(τL ε 1/2) for τ∈R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator L 0 in the norm of operators acting from the Sobolev space H s (with a suitable s) to L 2. Here L 0 is the effective operator with constant coefficients. Also, an approximation with corrector in the (H s→H 1)-norm for the operator L ε −1/2sin⁡(τL ε 1/2) 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 non-stationary Maxwell system in the case where the magnetic permeability is equal to μ, and the dielectric permittivity is given by the matrix η(x/ε).

KW - Homogenization

KW - Non-stationary Maxwell system

KW - Operator error estimates

KW - Periodic differential operators

KW - SPECTRAL APPROACH

KW - APPROXIMATION

KW - BLOCH-WAVE HOMOGENIZATION

KW - EQUATION

UR - http://www.scopus.com/inward/record.url?scp=85118853943&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/c84dab3a-fffd-32a6-80b9-ceeb898632f0/

U2 - 10.1016/j.jde.2021.10.054

DO - 10.1016/j.jde.2021.10.054

M3 - Article

VL - 307

SP - 348

EP - 388

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -