On the simplicity of the sloshing eigenvalues

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.