ASYMPTOTIC ANALYSIS OF A MULTIDIMENSIONAL VIBRATING STRUCTURE

The aim of this paper is to describe the qualitative behavior of the eigenfrequencies and eigenmotions of a model problem that represents the vibrations of an elastic multidimensional body (or multistructure). The model studied here assumes that the multidimensional structure consists of two bodies: one of them is a bounded domain of R(N) (N= 2 or 3 in practice), and the other a one-dimensional straight string (that is represented by a real interval). The bodies are elastically attached at a small neighborhood of a point of contact A on the boundary of the N-dimensional domain by one extreme of the string. When this structure undergoes impulses, both its parts vibrate. The result is the classical spectral problem for the Laplace operator in both regions of the multistructure, coupled with a special boundary condition, which models the junction between both bodies. It is a nonstandard eigenvalue system since the spectral problems corresponding to each part are linked through this junction condition. For a variety of reasons, there is interest in cases in which the junction region is very small. Thus one of the aims in this article is to study the asymptotic behavior of the spectrum of this eigenvalue problem when the junction region tends to disappear, and converges towards a set of Lebesgue measure zero containing the contact point. This is done in terms of the convergence of the Green's operator and the spectral family associated with this problem.