Spectral convergence for vibrating systems containing a part with negligible mass

We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domain Ω R of ℝ n , with n =2,3, which contains a fixed bounded domain B where the density takes the value 1 and 0 outside. Ω R has a diameter depending on a parameter R , with R ⩾1, diam(Ω R ) →∞ as R →∞ and the union of these sets is the whole space ℝ n (the half space { x ∈ℝ n / x n <0}, respectively). Depending on the dimension of the space n , and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R →∞. We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright © 2005 John Wiley & Sons, Ltd.