A spectral study of an infinite axisymmetric elastic layer

Lahcène Chorfi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 5, page 849-863
  • ISSN: 0764-583X

Abstract

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We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators A n , n , in a suitable Hilbert space. We show that the essential spectrum of A n is an interval of type [ γ , + [ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

How to cite

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Chorfi, Lahcène. "A spectral study of an infinite axisymmetric elastic layer." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.5 (2001): 849-863. <http://eudml.org/doc/194076>.

@article{Chorfi2001,
abstract = {We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators $A_n$, $n\in \mathbb \{N\}$, in a suitable Hilbert space. We show that the essential spectrum of $A_n$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.},
author = {Chorfi, Lahcène},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elasticity; axisymmetry; eigenmodes; min-max principle; axisymmetric layer; waves; spectral theory; eigenvalues; Hilbert spaces},
language = {eng},
number = {5},
pages = {849-863},
publisher = {EDP-Sciences},
title = {A spectral study of an infinite axisymmetric elastic layer},
url = {http://eudml.org/doc/194076},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Chorfi, Lahcène
TI - A spectral study of an infinite axisymmetric elastic layer
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 5
SP - 849
EP - 863
AB - We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators $A_n$, $n\in \mathbb {N}$, in a suitable Hilbert space. We show that the essential spectrum of $A_n$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.
LA - eng
KW - elasticity; axisymmetry; eigenmodes; min-max principle; axisymmetric layer; waves; spectral theory; eigenvalues; Hilbert spaces
UR - http://eudml.org/doc/194076
ER -

References

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  5. [0] D.J. Duterte, A.S. Bonnet-Ben Dhia and P. Joly, Mathematical analysis of elastic surface waves in topographic waveguides. M 3 AS (Math. Models Methods Appl. Sci.) 9 (1999) 755–798. Zbl0946.74034
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  8. [0] J. Miklowitz, The Theory of Elastic Waves and Wave Guides. North-Holland Publishing Company, Amsterdam, New York, Oxford (1980). Zbl0565.73025
  9. [0] J.A. Nitsche, On Korn’s second inequality. RAIRO Anal. Numér. 15 (1981) 237–248. Zbl0467.35019
  10. [0] B. Nkemzi and B. Heinrish, Partial Fourier approximation of the Lamé equation in axisymmetric domains. Math. Methods Appl. Sci. 22 (1999) 1017–1041. Zbl0932.65117
  11. [0] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New York, San Francisco, London (1978). Zbl0401.47001MR493421
  12. [0] M. Schechter, Operator Methods in Quantum Mechanics. North-Holland Publishing Company, Amsterdam, New York, Oxford (1981). Zbl0456.47012MR597895
  13. [0] G. A. Winbow, Seismic sources in open cased boreholes. Geophysics 56 (1991) 1040–1050. 

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