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On the simplicity of the sloshing eigenvalues

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia

Abstract

This paper investigates sloshing problems defined by $-Δu=0$ in $Ω$, with mixed boundary conditions: $\partial_νu=λu$ on $S$, and either $\partial_νu=0$ or $u=0$ on $W$. Here, $Ω$ represents a smooth bounded domain in $\mathbb{R}^n$ with boundary $\partialΩ=S \cup W$. We demonstrate that under small domain perturbations, all resulting eigenvalues are simple.

On the simplicity of the sloshing eigenvalues

Abstract

This paper investigates sloshing problems defined by in , with mixed boundary conditions: on , and either or on . Here, represents a smooth bounded domain in with boundary . We demonstrate that under small domain perturbations, all resulting eigenvalues are simple.
Paper Structure (11 sections, 10 theorems, 92 equations)

This paper contains 11 sections, 10 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Variational framework for Theorem \ref{['th:SD']}.
  3. Computation of derivatives
  4. No-splitting condition
  5. Perturbations supported in $W$
  6. Perturbations supported in $S$
  7. Proof of Theorem \ref{['th:SD']}
  8. Proof of Theorem \ref{['th:SN']}
  9. Perturbations supported in $S$
  10. Perturbations supported in $W$ (when $\Omega \subset \mathbb{R}^2$)
  11. A different variational approach to Theorem \ref{['th:SN']}.

Key Result

Theorem 1

For any $\varepsilon > 0$, there exists a perturbation $\psi \in \mathcal{D}$ with $\|\psi\| < \varepsilon$, leaving either $S$ or $W$ fixed, such that all eigenvalues of the Steklov-Dirichlet problem are simple.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • Lemma 7
  • proof
  • Remark 8
  • Lemma 9
  • ...and 10 more