On homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Issue 9 (3rd July 2020)

• Record Type: Journal Article
• Authors: Meshkova, Yu. M.
• Is Part Of: Applicable analysis. Volume 99:Issue 9(2020)
• Page Start: 1528
• Page End: 1563
• Publication Date: 2020-07-03
• Subjects: Periodic differential operators -- hyperbolic systems -- homogenization -- operator error estimates; 35B27 -- 35L53; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1540038
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: Let O ⊂ R d be a bounded domain of class C 3, 1 . In L 2 ( O ; C n ), we consider a self-adjoint matrix strongly elliptic second-order differential operator B D, ε, 0 < ε ⩽ 1, with the Dirichlet boundary condition. The coefficients of the operator B D, ε are periodic and depend on x / ε . We are interested in the behavior of the operators cos ⁡ ( t B D, ε 1 / 2 ) and B D, ε − 1 / 2 sin ⁡ ( t B D, ε 1 / 2 ), t ∈ R, in the small period limit. For these operators, approximations in the norm of operators acting from a certain subspace H of the Sobolev space H 4 ( O ; C n ) to L 2 ( O ; C n ) are found. Moreover, for B D, ε − 1 / 2 sin ⁡ ( t B D, ε 1 / 2 ), the approximation with the corrector in the norm of operators acting from H ⊂ H 4 ( O ; C n ) to H 1 ( O ; C n ) is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation ∂ t 2 u ε = − B D, ε u ε .