Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses

  • E. Sanchez-Palencia
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 195)


We study two physical problems containing a small parameter ε. When ε ↓0 there are infinitely many eigenvalues converging to zero. The corresponding asymptotic behavior is studied by a dilatation of the spectral plane. On the other hand, as ε ↓0, there are other eigenvalues converging to finite non-zero values. The first problem is the vibration of a thermoelastic bounded body where ε denotes the thermal conductivity. For ε = 0 the spectrum is formed by purely imaginary eigenvalues with finite multiplicity and the origin, which is an eigenvalue with infinite multiplicity ; for ε > 0 it becomes a set of eigenvalues with finite multiplicity. The second problem concerns the wave equation in dimension 3 with a distribution of density depending on ε, which converges, as ε ↓0 to a uniform density plus a punctual mass at the origin. As ε ↓0, there are “local vibrations” near the origin which are associated with the small eigenvalues.


Eigenvalue Problem Dirichlet Boundary Condition Small Eigenvalue Concentrate Mass Unbounded 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    AURIAULT J.L. et SANCHEZ-PALENCIA E. “Etude du comportement macroscopique d'un milieu poreux saturé d6formable”. Jour. Méca., 16 p. 575–603 (1977).MATHGoogle Scholar
  2. (2).
    KATO T. “Perturbation theory for Linear Operators”. Springer, Berlin (1966).CrossRefMATHGoogle Scholar
  3. (3).
    LADYZHENSRAYA O.A. “The Mathematical Theory of Viscous Incompressible Flow”. Gordon and Breach, New-York (1963).Google Scholar
  4. (4).
    OHAYON R. Personal communication (June 1983).Google Scholar
  5. (5).
    SANCHEZ-PALENCIA E. “tNon Homogeneous Media and Vibration Theory”. Springer, Berlin (1980).MATHGoogle Scholar
  6. (6).
    STEINBERG S. “Meromorphic Families of Compact Operators”. Arch. Rat. Mech. Anal. 31, p. 372–378 (1968). *** DIRECT SUPPORT *** A3418152 00008CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • E. Sanchez-Palencia
    • 1
  1. 1.Laboratoire de Mécanique Théorique, LA 229Université Paris VIParis Cedex 05France

Personalised recommendations