TY - JOUR

T1 - Homogenization of a nonstationary model equation of electrodynamics

AU - Dorodnyi, M. A.

AU - Suslina, T. A.

N1 - Dorodnyi, M.A., Suslina, T.A. Homogenization of a nonstationary model equation of electrodynamics. Math Notes 102, 645–663 (2017). https://doi.org/10.1134/S0001434617110050

PY - 2017

Y1 - 2017

N2 - In L 2(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 1/2ε) and ℒ −1/2ε sin(τℒ 1/2ε) 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 ∂ 2τ v ε = −ℒ ε v 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).

AB - In L 2(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 1/2ε) and ℒ −1/2ε sin(τℒ 1/2ε) 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 ∂ 2τ v ε = −ℒ ε v 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).

KW - periodic differential operator

KW - homogenization

KW - operator error estimate

KW - nonstationary Maxwell system

UR - https://link.springer.com/article/10.1134/S0001434617110050

M3 - Article

VL - 102

SP - 645

EP - 663

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 5

ER -