The bottom of the spectrum in a double-contrast periodic model

N Babych

doi:10.1007/978-0-8176-4899-2-6

The bottom of the spectrum in a double-contrast periodic model

N Babych

Department of Mathematical Sciences

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

Abstract

A periodic spectral problem in a bounded domain with double inhomogeneities in mass density and stiffness coefficients is considered. A previous study [BKS08] has explored the problem by the method of asymptotic expansions with justification of errors showing that all eigenelements of the homogenized problem really approximate some of the perturbed eigenelements. Within this chapter additional results, partly announced in [BKS08], are obtained on the eigenfunction convergence at the bottom of the spectrum. It is shown that the eigenfunctions, which correspond to the eigenvalues at the bottom of the spectrum, could converge either to zero or to the eigenfunctions of the homogenized problem. The result was obtained by the method of two-scale convergence [Al92, Zh00].

Original language	English
Title of host publication	Integral Methods in Science and Engineering, Volume 1
Subtitle of host publication	Analytic Methods
Editors	C Constanda, M E Perez
Place of Publication	Boston
Publisher	Birkhauser
Pages	53-63
Number of pages	11
Volume	1
ISBN (Electronic)	9780817648992
ISBN (Print)	9780817648985
DOIs	https://doi.org/10.1007/978-0-8176-4899-2-6
Publication status	Published - 2010
Event	10th International Conference on Integral Methods in Science and Engineering, IMSE 2008 - Santander, Spain Duration: 7 Jul 2010 → 10 Jul 2010

Conference

Conference	10th International Conference on Integral Methods in Science and Engineering, IMSE 2008
Country/Territory	Spain
City	Santander
Period	7/07/10 → 10/07/10

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10.1007/978-0-8176-4899-2-6

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Babych, N 2010, The bottom of the spectrum in a double-contrast periodic model. in C Constanda & ME Perez (eds), Integral Methods in Science and Engineering, Volume 1: Analytic Methods. vol. 1, Birkhauser, Boston, pp. 53-63, 10th International Conference on Integral Methods in Science and Engineering, IMSE 2008, Santander, Spain, 7/07/10. https://doi.org/10.1007/978-0-8176-4899-2-6

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abstract = "A periodic spectral problem in a bounded domain with double inhomogeneities in mass density and stiffness coefficients is considered. A previous study [BKS08] has explored the problem by the method of asymptotic expansions with justification of errors showing that all eigenelements of the homogenized problem really approximate some of the perturbed eigenelements. Within this chapter additional results, partly announced in [BKS08], are obtained on the eigenfunction convergence at the bottom of the spectrum. It is shown that the eigenfunctions, which correspond to the eigenvalues at the bottom of the spectrum, could converge either to zero or to the eigenfunctions of the homogenized problem. The result was obtained by the method of two-scale convergence [Al92, Zh00].",

author = "N Babych",

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language = "English",

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booktitle = "Integral Methods in Science and Engineering, Volume 1",

publisher = "Birkhauser",

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AU - Babych, N

PY - 2010

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N2 - A periodic spectral problem in a bounded domain with double inhomogeneities in mass density and stiffness coefficients is considered. A previous study [BKS08] has explored the problem by the method of asymptotic expansions with justification of errors showing that all eigenelements of the homogenized problem really approximate some of the perturbed eigenelements. Within this chapter additional results, partly announced in [BKS08], are obtained on the eigenfunction convergence at the bottom of the spectrum. It is shown that the eigenfunctions, which correspond to the eigenvalues at the bottom of the spectrum, could converge either to zero or to the eigenfunctions of the homogenized problem. The result was obtained by the method of two-scale convergence [Al92, Zh00].

AB - A periodic spectral problem in a bounded domain with double inhomogeneities in mass density and stiffness coefficients is considered. A previous study [BKS08] has explored the problem by the method of asymptotic expansions with justification of errors showing that all eigenelements of the homogenized problem really approximate some of the perturbed eigenelements. Within this chapter additional results, partly announced in [BKS08], are obtained on the eigenfunction convergence at the bottom of the spectrum. It is shown that the eigenfunctions, which correspond to the eigenvalues at the bottom of the spectrum, could converge either to zero or to the eigenfunctions of the homogenized problem. The result was obtained by the method of two-scale convergence [Al92, Zh00].

UR - http://link.springer.com/chapter/10.1007/978-0-8176-4899-2_6

U2 - 10.1007/978-0-8176-4899-2-6

DO - 10.1007/978-0-8176-4899-2-6

M3 - Chapter in a published conference proceeding

SN - 9780817648985

VL - 1

SP - 53

EP - 63

BT - Integral Methods in Science and Engineering, Volume 1

A2 - Constanda, C

A2 - Perez, M E

PB - Birkhauser

CY - Boston

T2 - 10th International Conference on Integral Methods in Science and Engineering, IMSE 2008

Y2 - 7 July 2010 through 10 July 2010

ER -

The bottom of the spectrum in a double-contrast periodic model

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