Bifurcating domains for an overdetermined eigenvalue problem in cylinders

Abstract

We study an overdetermined eigenvalue problem for domains $\hat{I}\text{c◯}$ contained in the half-cylinder $\hat{I}\pounds =\ddot{I}‰\times (0,+\infty )$, based on a bounded regular domain $\ddot{I}‰\subset {\mathbb{R}}^{N-1}$. It is easy to see that in any bounded cylinder $\hat{I}{\text{c◯}}_t=\ddot{I}‰\times (0,t)$, $t>0$, 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 $\hat{I}\text{c◯}\subset \hat{I}\pounds$ 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 $\hat{I}{\text{c◯}}_{t_j}$ at the values $t_j=\frac{\ddot{I}}{€}2\sqrt{\ddot{I}{ƒ}_j}$ where $\ddot{I}{ƒ}_j$ ($j\ge 1$) is a simple Neumann eigenvalue of the Laplace operator on $\ddot{I}‰\subset {\mathbb{R}}^{N-1}$. The solutions can be reflected with respect to $\ddot{I}‰$ to generate nontrivial solutions in a cylinder.