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

Abstract

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.