VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

Abstract

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending 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 order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.