Asymptotic behavior and spectral distortion for biharmonic Steklov problems on thin domains

Abstract

In this paper, we investigate the asymptotic behavior of the eigenvalues and eigenfunctions of a biharmonic Steklov problem defined on a thin domain in the $n$ dimensional Euclidean space degenerating to a segment. For $n=2$ the problem models the vibrations of a thin elastic plate with cross section represented by the given domain and mass concentrated on a free boundary. The problem under consideration depends on a parameter $\sigma$ that in the theory of elastic plates represents the Poisson ratio of the material. Our analysis points out a distortion in the limiting problem depending on $\sigma$ and the space dimension $n$.