TY - JOUR

T1 - Spectral approach to homogenization of hyperbolic equations with periodic coefficients

AU - Дородный, Марк Александрович

AU - Суслина, Татьяна Александровна

PY - 2018/6/15

Y1 - 2018/6/15

N2 - In L 2(R d;C n), we consider selfadjoint strongly elliptic second order differential operators A ε with periodic coefficients depending on x/ε ε>0. We study the behavior of the operators cos⁡(A ε 1/2τ) and A ε −1/2sin⁡(A ε 1/2τ), τ∈R, for small ε. Approximations for these operators in the (H s→L 2)-operator norm with a suitable s are obtained. The results are used to study the behavior of the solution v ε of the Cauchy problem for the hyperbolic equation ∂ τ 2v ε=−A εv ε+F. General results are applied to the acoustics equation and the system of elasticity theory.

AB - In L 2(R d;C n), we consider selfadjoint strongly elliptic second order differential operators A ε with periodic coefficients depending on x/ε ε>0. We study the behavior of the operators cos⁡(A ε 1/2τ) and A ε −1/2sin⁡(A ε 1/2τ), τ∈R, for small ε. Approximations for these operators in the (H s→L 2)-operator norm with a suitable s are obtained. The results are used to study the behavior of the solution v ε of the Cauchy problem for the hyperbolic equation ∂ τ 2v ε=−A εv ε+F. General results are applied to the acoustics equation and the system of elasticity theory.

KW - Effective operator

KW - Homogenization

KW - Hyperbolic equations

KW - Operator error estimates

KW - Periodic differential operators

UR - http://www.scopus.com/inward/record.url?scp=85042395446&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.02.023

DO - 10.1016/j.jde.2018.02.023

M3 - Article

VL - 264

SP - 7463

EP - 7522

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -