ON VIBRATING MEMBRANES WITH VERY HEAVY THIN INCLUSIONS

Abstract

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain Ω of ℝ2. The inclusion occupies a layer-like domain ωε of width 2ε and it has a density of order O(ε-m). The density is of order O(1) outside this inclusion ωε, the concentrated mass around the curve γ. ε and m are positive parameters, ε∈(0,1) and m>2. We set m=3 and show that low, middle and high frequency vibrations are necessary in order to describe the asymptotic behavior of the vibrations of the whole membrane. We study the asymptotic behavior, as ε→0, of these frequencies and of the corresponding eigenfunctions.