VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

We consider the asymptotic behavior of the vibrations of a membrane oc cupying a domain main Omega subset of R(2). The density, which depends on a small parameter epsilon, is of order O(1) out of certain regions where it is O(epsilon(-m)) with m > 0. These regions, the concentrate d masses with diameter O(epsilon), are located near the boundary, at m utual distances O(eta), with eta = eta(epsilon) --> 0. We impose Diric hlet (respectively Neumann) conditions at the points of partial deriva tive Omega in contact with (respectively, out of) the masses. Dependin g on the value of the parameter m (m > 2, m = 2 or m < 2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of orde r O(epsilon(m-2)) for m > 2, are approached via those of a local probl em obtained from the micro-structure of the problem, while the eigenva lues of order O(1) are approached through those of a homogenized probl em, which depend on the relation between epsilon and eta. Techniques o f boundary homogenization and spectral perturbation theory are used to study this problem.