On a vibrating plate with a concentrated mass

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter epsilon. The density is of order O (epsilon (-m)) in the small region, the concentrated mass, and it is of ord er O(1) outside; m is a positive parameter. Depending on the value of m (m < 2, m = 2 and m > 2) we describe the asymptotic behaviour, as epsilon --> 0, of the eigenvalues and eigenfunctions of the corresponding spectral prob lem. For m > 2 the vibrations associated with the low frequencies affect as ymptotically only a neighbourhood of the concentrated mass; we also conside r the asymptotic behaviour of the eigenfunctions associated with the high f requencies. (C) 2000 Academie des sciences/Editions scientifiques et medica les Elsevier SAS.