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On the Eigenvalues of the Biharmonic Steklov Problem on a Thin Set

Bauyrzhan Derbissaly, Nurbek kakharman

Abstract

This paper investigates the asymptotic behavior of the eigenvalues of the biharmonic operator on a thin set with Steklov boundary condition. The thin set is taken to be a tubular neighborhood of a planar smooth domain. We show that, as the thickness of this neighborhood tends to zero, all eigenvalues of the biharmonic operator with Steklov boundary condition converge to zero.

On the Eigenvalues of the Biharmonic Steklov Problem on a Thin Set

Abstract

This paper investigates the asymptotic behavior of the eigenvalues of the biharmonic operator on a thin set with Steklov boundary condition. The thin set is taken to be a tubular neighborhood of a planar smooth domain. We show that, as the thickness of this neighborhood tends to zero, all eigenvalues of the biharmonic operator with Steklov boundary condition converge to zero.
Paper Structure (10 sections, 4 theorems, 92 equations)

This paper contains 10 sections, 4 theorems, 92 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Preliminaries and notation
  3. The functional-geometric setting
  4. Modified problems
  5. Convergence of operators and their spectra
  6. Proof of Theorem \ref{['thm:1.1']}
  7. A key coercivity estimate
  8. Derivation of the limiting problem
  9. Proof of the compact convergence
  10. Spectral convergence: asymptotics of eigenvalues

Key Result

Theorem 1.1

Let $\{\lambda_{\varepsilon,k}\}_{k\in\mathbb{N}^+}$ be the sequence of eigenvalues of problem 2. Then, as $\varepsilon\to0$, where $\lambda_k$ is the $k$‐th eigenvalue of the following one‐dimensional problem with unknown $u=u(s)$ and eigenvalue $\lambda$. Here $s$ denotes the arclength parameter on $\partial\Omega$, and $\kappa(s)$ is the curvature at the point $s\in(0,|\partial\Omega|)$. More

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 3.1
  • proof
  • Remark 3.2