Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$ is a ball or an rectangle, we also know that $\int_{\partial \Omega} v \, ds=0$, where $ds$ is the length measure, due to exact formulas for the eigenfunctions on special domains. My question is that, is there any result on the classifications of domain $\Omega$ such that $\int_{\partial \Omega} v\, ds=0$ for any first Neumann Laplacian eigenfunction?
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