Bloch wave homogenization and spectral asymptotic analysis
type: Item
instance of: scholarly article
title: Bloch wave homogenization and spectral asymptotic analysis
zbMATH Open document ID: 0901.35005
DOI: 10.1016/S0021-7824(98)80068-8
author: Grégoire Allaire
author: Carlos Conca
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1998-11-12T00:00:00Z
review text: The authors consider a second order elliptic equation in a bounded periodically heterogeneous medium and the asymptotic behavior of its eigenvalues \((\lambda^k_{\varepsilon})^{-1}\) is studied as the structure period \(\varepsilon\) tends to zero. More precisely, the asymptotic of renormalized eigenvalues \(\varepsilon^{-2} \lambda^k_{\varepsilon}\), i.e. high frequency asymptotic, is considered. It is proved that this renormalized limit spectrum is devided into two parts: the so-called Bloch spectrum and the so-called boundary layer spectrum. The first one is defined explicitly as the spectrum of a family of new auxiliary problems, while the second corresponds to eigenfunctions concentrating at the boundary of the domain. In the case of a rectangular domain made of an entire periodicity cell, it is given an explicit description of the boundary layer spectrum.
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1187608
zbMATH Keywords: high frequency asymptotic
zbMATH Keywords: Bloch spectrum
zbMATH Keywords: boundary layer spectrum
MaRDI profile type: Publication
cites work: Eigenfrequencies of a tube bundle immersed in a fluid
cites work: Homogenization and Two-Scale Convergence
cites work: Multiscale convergence and reiterated homogenisation
cites work: Q4862181
cites work: Q4864326
cites work: Bloch-wave homogenization for a spectral problem in fluid-solid structures
cites work: Solution of Interface Problems by Homogenization. III
cites work: Q4692971
cites work: Q4197852
cites work: Sulla convergenza delle soluzioni di disequazioni variazionali
cites work: Q5690774
cites work: Q4714355
cites work: Q5286434
cites work: Q4339228
cites work: A spectral problem arising in fluid-solid structures
cites work: Homogenization of Periodic Structures via Bloch Decomposition
cites work: An introduction to \(\Gamma\)-convergence
cites work: Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. I. Scalar Model
cites work: Q5796198
cites work: Q3985317
cites work: A mathematical approach to the effective Hamiltonian in perturbed periodic problems
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: Q3996839
cites work: Q5526485
cites work: Homogenization of elliptic eigenvalue problems. II
cites work: Q3331073
cites work: On Wigner measures
cites work: A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential
cites work: An Approach for Constructing Families of Homogenized Equations for Periodic Media. II: Properties of the Kernel
cites work: First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof
cites work: A General Convergence Result for a Functional Related to the Theory of Homogenization
cites work: Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals
cites work: Q4734338
cites work: Eigenfrequencies of a tube bundle placed in a confined fluid
cites work: Global behaviour of large elastic tube bundles immersed in a fluid
cites work: Q4185363
cites work: Singular perturbations with nonsmooth limit and finite element approximation of layers for model problems of shells
cites work: Non-homogeneous media and vibration theory
cites work: A Dispersive Effective Medium for Wave Propagation in Periodic Composites
cites work: First-Order Corrections to the Homogenized Eigenvalues of a Periodic Composite Medium
cites work: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations
cites work: Homogenization of eigenvalue problems in perforated domains
cites work: Theory of Bloch waves
description: scientific article; zbMATH DE number 1187608