Comptes Rendus
no. 5
Asymptotically sharp uniform estimates in a scalar spectral stiff problem
[Estimations asymptotiques uniformes pour le spectre d'un problème scalaire raide]

Présenté par : Évariste Sanchez-Palencia

Miguel Lobo1 ; Serguei A. Nazarov2 ; Eugenia Pérez3 
1 Departamento de Matemáticas, Estadı́stica y Computación, Universidad de Cantabria, Av. de los Castros, s/n. 39005 Santander, Spain
2 Institute of Mechanical Engineering Problems, RAN V. O. Bol'shoi pr., 61, 199178 St Petersburg, Russia
3 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Av. de los Castros, s/n. 39005 Santander, Spain
Comptes Rendus. Mécanique, Volume 331 (2003) no. 5, pp. 325-330.

On obtienent des estimations de la vitesse de convergence des valeurs propres, convenablement mises a l'échelle, d'un problème de Neumann raide pour l'operateur de Laplace. Des bornes correspondantes a ces estimations sont exprimées en termes du rapport des raideurs et des propriétés du spectre limite dans les rangs des fréquences basses et moyennes.

Estimates of convergence rates for rescaled eigenvalues of the stiff Neumann problem for the Laplacian are obtained. The bounds are expressed in terms of the stiffness ratio and properties of the limit spectrum both for low and middle frequency ranges.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-0721(03)00073-1
Keywords: Acoustics, Waves, Vibrations, Stiff problem, Spectral analysis, Low frequencies, Middle frequencies
Mot clés : Acoustique, Ondes, Vibrations, Problème raide, Analyse spectrale, Basses fréquences, Moyennes fréquences
Miguel Lobo 1 ; Serguei A. Nazarov 2 ; Eugenia Pérez 3

1 Departamento de Matemáticas, Estadı́stica y Computación, Universidad de Cantabria, Av. de los Castros, s/n. 39005 Santander, Spain
2 Institute of Mechanical Engineering Problems, RAN V. O. Bol'shoi pr., 61, 199178 St Petersburg, Russia
3 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Av. de los Castros, s/n. 39005 Santander, Spain 
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     author = {Miguel Lobo and Serguei A. Nazarov and Eugenia P\'erez},
     title = {Asymptotically sharp uniform estimates in a scalar spectral stiff problem},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {325--330},
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Miguel Lobo; Serguei A. Nazarov; Eugenia Pérez. Asymptotically sharp uniform estimates in a scalar spectral stiff problem. Comptes Rendus. Mécanique, Volume 331 (2003) no. 5, pp. 325-330. doi : 10.1016/S1631-0721(03)00073-1. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00073-1/

[1] M. Lobo; É. Sanchez-Palencia Low and high frequency vibration in stiff problems, De Giorgi 60th birthday. Partial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser, Boston, 1989, pp. 729-742

[2] M. Lobo; E. Pérez High frequency viabrations in a stiff problem, Math. Models Methods Appl. Sci., Volume 7 (1997) no. 2, pp. 291-311

[3] G.P. Panasenko Asymptotics of solutions and eigenvalues of elliptic equations with strongly variable coefficients, Dokl. Akad. Nauk, Volume 252 (1980) no. 6, pp. 1320-1325 English transl.: Soviet Math. Dokl. 21 (1980) 942–947

[4] G.P. Panasenko Asymptotics of the eigenvalues of elliptic equations with strongly varying coefficients, Trudy Sem. Petrovsk., Volume 12 (1987), pp. 201-217

[5] M. Lobo, S.A. Nazarov, E. Pérez, Eigenoscillations of contrastly non-homogeneous elastic body. Asymptotic and uniform estimates for the eigenvalues, in preparation

[6] É. Sanchez-Palencia Asymptotic and spectral properites of a class of singular-stiff problems, J. Math. Pures Appl., Volume 71 (1992), pp. 379-406

[7] J. Sanchez-Hubert; É. Sanchez-Palencia Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer-Verlag, Heidelberg, 1989

[8] S.A. Nazarov Asymptotic Theory of Thin Plates and Rods. Vol. 1. Dimension Reduction and Integral Estimates, Nauchnaya Kniga, Novosibirck, 2002

[9] S.A. Nazarov Estimating the convergence rate for eigenfrequences of anisotropic plates with variable thickness, C. R. Mecanique, Volume 330 (2002), pp. 603-607

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