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

Bauyrzhan Derbissaly, Pier Domenico Lamberti

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 $σ$ 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 $σ$ and the space dimension $n$.

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 dimensional Euclidean space degenerating to a segment. For 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 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 and the space dimension .
Paper Structure (5 sections, 4 theorems, 85 equations)

This paper contains 5 sections, 4 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Proof of Theorem \ref{['thm 1.1']}
  4. Finding the limiting problem
  5. Spectral convergence results

Key Result

Theorem 1.1

Let $n\geq2$. Then for all $k\in \mathbb{N}$, where $\lambda_k$ are the eigenvalues of the 4th-order Sturm-Liouville problem and $\mathcal{N}=(n-1)/(1-2\sigma+\sigma n)$. Moreover, there exists an orthonormal basis of eigenfunctions $v_k$, $k\in \mathbb{N}$ of 1.4 in $L^2_{n,\rho}(-l,l)$ such that, possibly passing to a subsequence, as $\epsilon\to0$, where $v_k$ is constantly extended in the v

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof