ON THE LOCAL VIBRATIONS FOR SYSTEMS WITH MANY CONCENTRATED MASSES

We consider the asymptotic behaviour of the vibrations of a body occup ying a domain Omega subset of R-n, n = 2, 3. The density, which depend s on a small parameter epsilon, is of the order O(1) out of certain re gions where it is O(epsilon(-m)) with m > 2. These regions, the concen trated masses with diameter O(epsilon), are located near the boundary, at mutual distances O(eta), with II = eta(epsilon) --> 0. We impose D irichlet (respectively, Neumann) conditions at the points of partial d erivative Omega in contact with (respectively, out of) the masses. Som e of the eigenvalues, small eigenvalues of order O(epsilon(m-2)), are approached by those of a local problem obtained from the microstructur e of the problem. We describe here the structure of the corresponding eigenfunctions. We also give a result dealing with the multiplicity of these eigenvalues.