On the simplicity of the sloshing eigenvalues

TL;DR

This paper studies the eigenvalues of sloshing problems in smooth bounded domains, showing that small domain perturbations lead to simple eigenvalues, which simplifies the spectral analysis of such physical systems.

Contribution

It proves that eigenvalues remain simple under small perturbations of the domain boundary in sloshing problems with mixed boundary conditions.

Findings

Eigenvalues are simple under small domain perturbations.

The analysis applies to mixed boundary conditions in bounded domains.

Results facilitate understanding of spectral stability in fluid sloshing models.

Abstract

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