Low and High Frequency Vibration in Stiff Problems

Abstract

Stiff problems are partial differential equation problems with very different values of the coefficients in two regions of the domain Ω of the space variables. Very different means that the ratio of coefficients is described by a parameter ∈ → 0. Stiff problems originated in Lions [4]. The behavior of the corresponding eigenvalue problems depends highly on the form of the H-like space (in the standard framework V ⊂ H with dense and compact imbedding) and in particular on the fact that the corresponding coefficients do depend or not on ∈. We refer to Lions [5], Geymonat et al. [1,2], Panasenko [7], Gibert [3] and Sanchez-Palencia [8] for the study of several kinds of eigenvalue stiff problems. It turns out that the case when the coefficients in front of the H-like space do not depend on ∈ was not very widely explored, as the “low frequency” region does not provide a good insight on the vibration problems all over Ω. We consider here such problems, in particular the high frequency region. In order to obtain properties of the spectral family we use a Fourier transform technique starting from the solutions of the corresponding hyperbolic problems involving second order derivatives with respect to time. (We refer to Lobo and Sanchez Palencia [6] and Sanchez-Palencia [8], chap. 12, sect. 3 for other problems using this method).