M. Lobo et E. Perez, VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY, Mathematical models and methods in applied sciences, 5(5), 1995, pp. 565-585
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.