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

TL;DR

This paper studies the asymptotic behavior of eigenvalues and eigenfunctions for a biharmonic Steklov problem on thin domains, revealing how spectral properties are affected by domain degeneration, material parameters, and space dimension.

Contribution

It provides a detailed analysis of spectral distortion and asymptotics for biharmonic Steklov problems on thin domains, highlighting the influence of the Poisson ratio and dimension.

Findings

Eigenvalues and eigenfunctions exhibit specific asymptotic behavior as the domain degenerates.

Spectral distortion depends on the Poisson ratio and space dimension.

Results elucidate the impact of domain thinness on elastic plate vibrations.

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 $n$.