Bifurcating domains for an overdetermined eigenvalue problem in cylinders
Yuanyuan Lian, Filomena Pacella, Pieralberto Sicbaldi
TL;DR
The paper studies a relative overdetermined eigenvalue problem for domains inside a half-cylinder and constructs nontrivial domains supporting a positive eigenfunction by bifurcating from the trivial cylindrical domains at critical radii determined by simple Neumann eigenvalues on the base domain. It develops a formalism of mixed boundary problems, pulls the problem back to a fixed reference domain, and introduces a normal-derivative operator whose zeros correspond to overdetermined solutions. The main result is a Crandall–Rabinowitz bifurcation: near each simple eigenvalue, a smooth curve of nontrivial hypographs exists, yielding new domains with positive eigenfunctions satisfying the overdetermined conditions, and these solutions can be reflected to obtain cylinder-wide counterparts. This provides prototype bifurcation results for relative overdetermined problems in cylinders and suggests avenues for extending to semilinear settings and broader geometries.
Abstract
We study an overdetermined eigenvalue problem for domains $Ω$ contained in the half-cylinder $Σ=ω\times (0, +\infty)$, based on a bounded regular domain $ω\subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $Ω_{t}=ω\times (0, t)$, $t > 0$, 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 $Ω\subset Σ$ 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 $Ω_{t_j}$ at the values $t_{j} = \fracπ{2\sqrt{σ_j}}$ where $σ_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $ω\subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $ω$ to generate nontrivial solutions in a cylinder.