ASYMPTOTIC ANALYSIS OF A MULTIDIMENSIONAL VIBRATING STRUCTURE

The aim of this paper is to describe the qualitative behavior of the e igenfrequencies and eigenmotions of a model problem that represents th e vibrations of an elastic multidimensional body (or multistructure). The model studied here assumes that the multidimensional structure con sists 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 attache d at a small neighborhood of a point of contact A on the boundary of t he N-dimensional domain by one extreme of the string. When this struct ure undergoes impulses, both its parts vibrate. The result is the clas sical spectral problem for the Laplace operator in both regions of the multistructure, coupled with a special boundary condition, which mode ls the junction between both bodies. It is a nonstandard eigenvalue sy stem since the spectral problems corresponding to each part are linked through this junction condition. For a variety of reasons, there is i nterest 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 conta ining the contact point. This is done in terms of the convergence of t he Green's operator and the spectral family associated with this probl em.