Bifurcating domains for an overdetermined eigenvalue problem in cylinders

TL;DR

This paper constructs new overdetermined eigenvalue problem solutions in cylinders by bifurcating from trivial domains at specific eigenvalues, revealing complex domain structures beyond simple cylinders.

Contribution

It demonstrates the existence of bifurcating domains with positive eigenfunctions solving overdetermined problems, extending known solutions in cylindrical domains.

Findings

Branches of domains bifurcate from trivial cylinders at specific eigenvalues.

Constructed nontrivial solutions by reflecting bifurcating domains.

Identified bifurcation points related to Neumann eigenvalues of the cross-section.

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.