On the simplicity of the sloshing eigenvalues

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