On the simplicity of the sloshing eigenvalues

Abstract

This paper investigates sloshing problems defined by $-\hat{I}”u=0$ in $\hat{I}\text{c◯}$, with mixed boundary conditions: ${\partial }_{\hat{I}}½u=\hat{I}»u$ on $S$, and either ${\partial }_{\hat{I}}½u=0$ or $u=0$ on $W$. Here, $\hat{I}\text{c◯}$ represents a smooth bounded domain in ${\mathbb{R}}^n$ with boundary $\partial \hat{I}\text{c◯}=S\cup W$. We demonstrate that under small domain perturbations, all resulting eigenvalues are simple.