An overdetermined problem for sign-changing eigenfunctions in unbounded domains

TL;DR

This paper constructs unbounded, periodic domains in the plane where a specific PDE with sign-changing eigenfunctions admits solutions, revealing new geometric configurations for such spectral problems.

Contribution

It introduces a method to construct unbounded domains with periodic structures where sign-changing solutions to a PDE exist, expanding understanding of eigenfunctions in unbounded regions.

Findings

Existence of unbounded domains with periodic solutions

Construction of domains bifurcating from strips

Solutions change sign and satisfy boundary conditions

Abstract

We study the existence of non-trivial unbounded domains of $\Omega \subset \mathbb{R}^2$ where the equation \begin{align} - \lambda u_{xx} -u_{tt} &= u \qquad \text{in $\Omega$,}\nonumber u &=0 \qquad \text{on $\partial \Omega$,}\nonumber \end{align} is solvable subject to the conditions \begin{align} \frac{\partial u}{\partial \eta} =-1\quad \text{on $\partial \Omega^+$} \quad \textrm{and}\quad \frac{\partial u}{\partial \eta} =+1\quad \text{on $\partial \Omega^-$.} \end{align} For every integer $m\geq 0$, we prove the existence of a family of unbounded domains $\Omega\subset \mathbb{R}^2$ indexed by $0 \leqslant\ell\leqslant 2m$, where the above problem admits periodic sign-changing solutions. The domains we construct are periodic in the first coordinate in $\mathbb{R}^2$, and they bifurcate from suitable strips.