Bifurcating domains for an overdetermined eigenvalue problem in cylinders

Abstract

We study an overdetermined eigenvalue problem for domains $\Omega$ contained in the half-cylinder $\Sigma=\omega \times (0, +\infty)$, based on a bounded regular domain $\omega \subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $\Omega_{t}=\omega \times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $\Omega\subset \Sigma$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $\Omega_{t_j}$ at the values $t_{j} = \frac{\pi}{2\sqrt{\sigma_j}}$ where $\sigma_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $\omega \subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $\omega$ to generate nontrivial solutions in a cylinder.