Payne's nodal line conjecture fails on doubly-connected planar domains

TL;DR

This work shows that Payne's nodal line conjecture for planar domains cannot hold beyond simply-connected regions by constructing a bounded doubly-connected planar domain with a second Dirichlet eigenfunction whose nodal line is closed and does not touch the boundary. The authors introduce the sliding-handle framework (FDSH), attaching a thin annulus to a base domain and tracking a two-parameter family of domains under a continuity- and symmetry-preserving perturbation; they prove that under this framework, a parameter value exists for which the nodal line is enclosed within the annulus, forming a closed Jordan curve. A key technical contribution is establishing the nodal line's controlled behavior under domain deformation, including a lower semicontinuity property and a topological argument that prevents simultaneous contact with both boundary components. The results imply the minimal hole-count for counterexamples can be as low as two boundary components, and the method provides a general mechanism to generate similar domains (even with more holes) by modifying the internal geometry without compromising the essential nodal-line trapping. Overall, the paper advances the understanding of nodal sets for the Laplacian and clarifies the topological limits of Payne's conjecture in the plane.

Abstract

We present examples of bounded planar domains with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne's nodal line conjecture can at most hold for simply-connected domains in the plane.

Figures (5)

• Figure 1: The family of doubly-connected domains described in the introduction, in different configurations. In each case, the dotted line is the segment bisecting the rectangle and the blue line is the expected profile of the nodal line.
• Figure 2: Two samples of an FDSH. The expected nodal line is in blue. The segment $\sigma$ along which the eigenfunction changes sign is in red.
• Figure 3: Three typical situations for the function $v$ of Lemma \ref{['lemme:E recouvrent T']}. The nodal line is drawn in blue, and $v$ changes sign along the red segment, at the point $P$. The green arc in (\ref{['fig:ligne_nodale_anneau_3']}) corresponds to the path $\gamma'$ constructed in the proof of the lemma.
• Figure 4: The domain $\Theta_{h,t}$ constructed in Lemma \ref{['lemme:famille à deux paramètres']} at $t=t_-$ and $t=t_+$, for some $0<h<h_0$. The segment $(W,E)$ is drawn in dark red, while its neighbourhood $\mathcal{U}$ is displayed in light red.
• Figure 5: The domain $\Theta_{h,t}$ constructed in the proof of Proposition \ref{['prop:definition de la famille de domaines']} at $t=t_-$ and $t=t_+$, for some $0<h<h_2$. The vertical strip in light blue represents the neighbourhood $\mathcal{V}$. The green region is an arbitrary compact set $K\subseteq\Omega$.

Theorems & Definitions (9)

• Conjecture : Nodal line conjecture 1967
• Proposition 2.4
• proof : Proof of Theorem \ref{['thm:resultat principal']}
• proof
• proof
• proof : Proof of Theorem \ref{['thm:ligne nodale famille doublement connexe']}
• proof
• proof : Proof of Proposition \ref{['prop:definition de la famille de domaines']}
• proof