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Singular perturbations of spectra

Part of the Lecture Notes in Mathematics book series (LNM,volume 711)

Abstract

A mathematical description of free vibrations of a membrane leads to eigenvalue problems for elliptic differential operators containing a small positive parameter ɛ in the highest order part. The asymptotic behaviour (for ɛ → +0) of the eigenvalues is studied in second order problems that reduce to zero-th and first order for ɛ=0 and in a fourth order problem that reduces to an elliptic problem of second order. In the case of reduction to zero-thorder the density of the eigenvalues on a half-axis grows beyond bound and is proportional to ɛ−n/2 (in n dimensions). In the case of reduction to first order the relation between the asymptotic behaviour of the spectrum and the critical points of the reduced operator is shown. In the case of reduction to second order an asymptotic series expansion is constructed for every eigenvalue.

Keywords

  • Singular Perturbation
  • Singular Perturbation Problem
  • Elliptic Differential Operator
  • Small Positive Parameter
  • Order Elliptic Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1979 Springer-Verlag

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de Groen, P.P.N. (1979). Singular perturbations of spectra. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062945

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  • DOI: https://doi.org/10.1007/BFb0062945

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09245-2

  • Online ISBN: 978-3-540-35332-4

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