TY - JOUR

T1 - Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain

AU - Suslina, T. A.

N1 - Suslina, T.A. Homogenization of the Stationary Maxwell System with Periodic Coefficients in a Bounded Domain. Arch Rational Mech Anal 234, 453–507 (2019). https://doi.org/10.1007/s00205-019-01394-5

PY - 2019

Y1 - 2019

N2 - In a bounded domain O⊂ R 3 of class C 1 , 1, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by η(x/ ε) and μ(x/ ε) , where η(x) and μ(x) are symmetric (3 × 3) -matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here ε> 0 is the small parameter. We use the following notation for the solutions of the Maxwell system: u ε and v ε are the electric and magnetic field intensities, w ε and z ε are the electric and magnetic displacement vectors. It is known that u ε, v ε, w ε, and z ε weakly converge in L 2(O) to the corresponding homogenized fields u, v, w, and z (the solutions of the homogenized Maxwell system with the effective coefficients), as ε→ 0. We improve the classical results and find approximations for u ε, v ε, w ε, and z ε in the L 2(O) -norm. The error terms do not exceed Cε(‖q‖L2+‖r‖L2), where the divergence free vector-valued functions q and r are the right-hand sides of the Maxwell equations.

AB - In a bounded domain O⊂ R 3 of class C 1 , 1, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by η(x/ ε) and μ(x/ ε) , where η(x) and μ(x) are symmetric (3 × 3) -matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here ε> 0 is the small parameter. We use the following notation for the solutions of the Maxwell system: u ε and v ε are the electric and magnetic field intensities, w ε and z ε are the electric and magnetic displacement vectors. It is known that u ε, v ε, w ε, and z ε weakly converge in L 2(O) to the corresponding homogenized fields u, v, w, and z (the solutions of the homogenized Maxwell system with the effective coefficients), as ε→ 0. We improve the classical results and find approximations for u ε, v ε, w ε, and z ε in the L 2(O) -norm. The error terms do not exceed Cε(‖q‖L2+‖r‖L2), where the divergence free vector-valued functions q and r are the right-hand sides of the Maxwell equations.

UR - http://www.scopus.com/inward/record.url?scp=85065469119&partnerID=8YFLogxK

U2 - 10.1007/s00205-019-01394-5

DO - 10.1007/s00205-019-01394-5

M3 - Article

VL - 234

SP - 453

EP - 507

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -