Vibrating systems with concentrated masses:: On the high frequencies and the local problem

We consider the vibrations of a membrane occupying a domain Omega of R-2, that contains many small regions of high density near the boundary, the concentrated masses. Considering rapidly alternating mixed boundary conditions, we study the asymptotic behavior, when epsilon --> 0, of the eigenelements (lambda(epsilon), u(epsilon)) of the corresponding spectral problem. We show that the computation of correcting terms for certain eigenfunctions u(epsilon) associated with the high frequencies of (2) is deeply involved with the study of the high frequencies of the microscopic or local problem.