On Different Quasimodes for the Homogenization of Steklov-Type Eigenvalue Problems

  • M. Lobo
  • M. E. Pérez


Roughly speaking, a quasimode for an operator with a discrete spectrum on a Hilbert space can be defined as a pair (\(\tilde{w}, \mu\)), where \(\tilde{w}\) is a function approaching a certain linear combination of eigenfunctions associated with the eigenvalues of the operator in a “small interval” [\(\mu - r, \mu + r\)]. The remainder r also deals with the discrepancies between \(\tilde{w}\) and the eigenfunctions.

The value of the quasimodes in describing asymptotics for low and high frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been made clear recently in many papers. We refer to [Pe08] for an abstract general framework that can be applied to several problems of spectral perturbation theory and to [LoPe03] and [SaSa89] for a large variety of these problems. As a matter of fact, for these problems, the spaces and the operators under consideration depend on the parameter of perturbation, and the function \(\tilde{w}\) and the numbers μ and r arising in the definition of a quasimode can also depend on this parameter.


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  1. [BuIo06].
    Bucur, D., Ionescu, I.: Asymptotic analysis and scaling of friction parameters. Z. Angew. Math. Phys., 57, 1–15 (2006).CrossRefMathSciNetGoogle Scholar
  2. [CaDa04].
    Campillo, M., Dascalu, C., Ionescu, I.: Instability of a periodic system of faults. Geophys. J. Internat., 159, 212–222 (2004).CrossRefGoogle Scholar
  3. [Gr92].
    Grisvard, P.: Singularities in Boundary Value Problems, Masson, Paris (1992).MATHGoogle Scholar
  4. [IoDa02].
    Ionescu, I., Dascalu, C., Campillo, M.: Slip-weakening friction on a periodic system of faults: spectral analysis. Z. Angew. Math. Phys., 53, 950–995 (2002).CrossRefMathSciNetGoogle Scholar
  5. [LoPe03].
    Lobo, M., Pérez, E.: Local problems in vibrating systems with concentrated masses: a review. C.R. Mécanique, 331, 303–317 (2003).MATHCrossRefGoogle Scholar
  6. [LoPe09].
    Lobo, M., Pérez, E.: Long time approximations for solutions of wave equations associated with Steklov spectral homogenization problems. (Submitted to referee.)Google Scholar
  7. [OlSh92].
    Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization, North-Holland, London (1992).Google Scholar
  8. [PaPe07].
    Panasenko, G.P., Pérez, E.: Asymptotic partial decomposition of domain for spectral problems in rod structures. J. Math. Pures Appl., 87, 1–36 (2007).MATHMathSciNetGoogle Scholar
  9. [Pe07].
    Pérez, E.: On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete Cont. Dyn. Syst. Ser. B, 7, 859–883 (2007).MATHGoogle Scholar
  10. [Pe08].
    Pérez, E.: Long time approximations for solutions of wave equations via standing waves from quasimodes. J. Math. Pures Appl., 90, 387–411 (2008).MATHMathSciNetGoogle Scholar
  11. [SaSa89].
    Sanchez-Hubert, J., Sanchez-Palencia, E: Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer, Heidelberg (1989).MATHGoogle Scholar

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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Universidad de CantabriaSantanderSpain

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