An overdetermined problem for sign-changing eigenfunctions in unbounded domains

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.