A spectral study of an infinite axisymmetric elastic layer
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 5, pp. 849-863.

Nous présentons ici une étude théorique des modes propres dans une couche élastique axisymétrique. La modélisation mathématique permet de ramener ce problème à l’étude spectrale d’une suite d’opérateurs A n , n, non bornés et autoadjoints dans un espace de Hilbert adéquat. On montre que le spectre essentiel de A n est un intervalle du type [γ,+[ et que, sous certaines conditions portant sur les coefficients du milieu, le spectre discret est non vide.

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.

Classification : 35P15, 47A70, 73D30
Mots clés : elasticity, axisymmetry, eigenmodes, min-max principle
@article{M2AN_2001__35_5_849_0,
     author = {Chorfi, Lahc\`ene},
     title = {A spectral study of an infinite axisymmetric elastic layer},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {849--863},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {5},
     year = {2001},
     mrnumber = {1866270},
     zbl = {0994.35100},
     language = {en},
     url = {http://archive.numdam.org/item/M2AN_2001__35_5_849_0/}
}
TY  - JOUR
AU  - Chorfi, Lahcène
TI  - A spectral study of an infinite axisymmetric elastic layer
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2001
SP  - 849
EP  - 863
VL  - 35
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/M2AN_2001__35_5_849_0/
LA  - en
ID  - M2AN_2001__35_5_849_0
ER  - 
%0 Journal Article
%A Chorfi, Lahcène
%T A spectral study of an infinite axisymmetric elastic layer
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2001
%P 849-863
%V 35
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/item/M2AN_2001__35_5_849_0/
%G en
%F M2AN_2001__35_5_849_0
Chorfi, Lahcène. A spectral study of an infinite axisymmetric elastic layer. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 5, pp. 849-863. http://archive.numdam.org/item/M2AN_2001__35_5_849_0/

[0] A. Bamberger, Y. Dermenjian and P. Joly, Mathematical analysis of the propagation of elastic guided waves in heterogeneous media. J. Differential Equations 88 (1990) 113-154. | Zbl

[0] A. Bamberger, P. Joly and M. Kern, Propagation of elastic surface waves along a cylindrical cavity of arbitrary cross section. RAIRO Modél. Math. Anal. Numér. 25 (1991) 1-30. | Numdam | Zbl

[0] M. Bouchon and D.P. Schmitt, Full-wave acoustic logging in an irregular borehole. Geophysics 54 (1989) 758-765.

[0] L. Chorfi, Étude mathématique des modes guidés dans un milieu élastique à symétrie de révolution. RAIRO Modél. Math. Anal. Numér. 30 (1996) 299-342. | Numdam | Zbl

[0] D.J. Duterte, A.S. Bonnet-Ben Dhia and P. Joly, Mathematical analysis of elastic surface waves in topographic waveguides 9 (1999) 755-798. | Zbl

[0] G. Duvaut, Mécanique des milieux continus. Masson, Paris (1990).

[0] T. Kato, Perturbation Theory for Linear Operators. 2nd edn., Springer-Verlag, New York (1976). | MR | Zbl

[0] J. Miklowitz, The Theory of Elastic Waves and Wave Guides. North-Holland Publishing Company, Amsterdam, New York, Oxford (1980). | Zbl

[0] J.A. Nitsche, On Korn's second inequality. RAIRO Anal. Numér. 15 (1981) 237-248. | Numdam | Zbl

[0] B. Nkemzi and B. Heinrish, Partial Fourier approximation of the Lamé equation in axisymmetric domains. Math. Methods Appl. Sci. 22 (1999) 1017-1041. | Zbl

[0] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New York, San Francisco, London (1978). | MR | Zbl

[0] M. Schechter, Operator Methods in Quantum Mechanics. North-Holland Publishing Company, Amsterdam, New York, Oxford (1981). | MR | Zbl

[0] G. A. Winbow, Seismic sources in open cased boreholes. Geophysics 56 (1991) 1040-1050.