Homogenization of the spectral Dirichlet problem for a system of differential equations with rapidly oscillating coefficients and changing sign density

被引：0

作者：

Nazarov S.A. ^{[1
]}

Pyatnitskii A.L. ^{[2
,3
]}

机构：

[1] Institute of Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg 199178, 61, Bolshoi pr. V.O.

[2] The Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119333, 53, Leninskiy pr.

[3] Narvik Institute of Technology

来源：

Journal of Mathematical Sciences | 2010年 / 169卷 / 2期

关键词：

Discrete Spectrum; Spectral Problem; Periodicity Cell; Integral Identity; Simple Eigenvalue;

D O I：

10.1007/s10958-010-0047-2

中图分类号：

学科分类号：

摘要：

We study the asymptotic behavior of the spectrum of the Dirichlet problem for a formally selfadjoint elliptic system of differential equations with rapidly oscillating coefficients and changing sign density ρ. Since the factor ρ at the spectral parameter changes sign, the problem possesses two - positive and negative - infinitely large sequences of eigenvalues. Their asymptotic structure essentially depends on whether the mean ρ̄ over the periodicity cell vanishes. In particular, in the case ρ̄ = 0, the homogenized problem becomes a quadratic pencil. Bibliography: 20 titles. © 2010 Springer Science+Business Media, Inc.

引用

页码：212 / 248

页数：36

共 19 条

[11] Zhikov V.V., Kozlov S.M., Oleinik O.A., Homogenization of Differential Operators and Integral Functions [in Russian], (1993)
[12] Vanninatan M., Homogenization of eigenvalue problems in perforated domains, Pro. Indian Acad. Sci. (Math. Sci.), 90, 3, pp. 239-271, (1981)
[13] Oleinik O.A., Shamaev A.S., Yosifyan G.A., Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media, (1990)
[14] Cardone G., Corbo Esposito A., Nazarov S.A., Korn's inequality for periodic solids and convergence rate of homogenization, Appl. Anal., 88, 6, pp. 847-876, (2009)
[15] Gelfand I.M., Shilov G.E., Generalized Functions and Operations over Them, (1958)
[16] Vishik M.I., Lusternik L.A., Regular degeneration and boundary layer for linear differential equations with small parameter, Uspekhi Mat. Nauk, 12, 5, pp. 3-122, (1957)
[17] Gohberg I.C., Krein M.G., Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Space [in Russian], (1965)
[18] Nazarov S.A., On the asymptotics of the spectrum of a thin plate problem of elasticity, Sib. Mat. Zh., 41, 4, pp. 895-912, (2000)
[19] Berdicevskii V.L., High-frequency longwave oscillations of plates, Dokl. Akad. Nauk SSSR, 236, 6, pp. 1319-1322, (1977)

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