Comptes Rendus
Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube
Comptes Rendus. Mécanique, Volume 331 (2003) no. 8, pp. 531-536.

We consider the eigenvalue problem for the Laplace operator in a bounded three-dimensional domain where a thin tube is cut out. Imposing a Neumann boundary condition on the boundary of this tube, we construct asymptotics for eigenvalues on the small parameter that is a diameter of the tube.

On considère l'opérateur de Laplace dans un domaine tridimensionnel borné dont on a extrait un tube fin, avec la condition aux limites de Neumann. Nous construisons le développement asymptotique des valeurs propres pour des valeurs petites du diamètre du tube.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-0721(03)00125-6
Keywords: Vibrations, Asymptotique, Valeur propre
Mot clés : Vibrations, Asymptotics, Eigenvalue

Marina Yu. Planida 1

1 The Bashkir State Pedagogical University, October Revolution st., 3a, 450000 Ufa, Russia
@article{CRMECA_2003__331_8_531_0,
     author = {Marina Yu. Planida},
     title = {Asymptotics for eigenvalues of the {Laplacian} with a {Neumann} boundary condition on a thin cut-out tube},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {531--536},
     publisher = {Elsevier},
     volume = {331},
     number = {8},
     year = {2003},
     doi = {10.1016/S1631-0721(03)00125-6},
     language = {en},
}
TY  - JOUR
AU  - Marina Yu. Planida
TI  - Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube
JO  - Comptes Rendus. Mécanique
PY  - 2003
SP  - 531
EP  - 536
VL  - 331
IS  - 8
PB  - Elsevier
DO  - 10.1016/S1631-0721(03)00125-6
LA  - en
ID  - CRMECA_2003__331_8_531_0
ER  - 
%0 Journal Article
%A Marina Yu. Planida
%T Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube
%J Comptes Rendus. Mécanique
%D 2003
%P 531-536
%V 331
%N 8
%I Elsevier
%R 10.1016/S1631-0721(03)00125-6
%G en
%F CRMECA_2003__331_8_531_0
Marina Yu. Planida. Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube. Comptes Rendus. Mécanique, Volume 331 (2003) no. 8, pp. 531-536. doi : 10.1016/S1631-0721(03)00125-6. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00125-6/

[1] M.Yu. Planida On the convergence of solutions of singularly perturbed boundary-value problems for the Laplace operator, Math. Notes, Volume 71 (2002), pp. 867-877

[2] A.M. Il'in Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, RI, 1992

[3] R.R. Gadyl'shin Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition, Math. Notes, Volume 52 (1992), pp. 1020-1029

[4] M.Yu. Planida Asymptotics of eigenvalues for a cylinder insulated on a narrow strip, Comput. Math. Math. Phys., Volume 43 (2003), pp. 403-413

[5] S.A. Nazarov, M.V. Paukshto, Discrete models and homogenization in problems of elastisity theory, Lenigrad University, Leningrad, 1984 (in Russian)

[6] S. Ozawa Spectra of domains spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 30 (1983), pp. 259-277

[7] V.G. Maz'ya; S.A. Nazarov; B.A. Plamenevskij Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Math. USSR-Izv., Volume 24 (1985), pp. 321-345

Cited by Sources:

Comments - Policy