Neumann's nodal line may be closed on doubly-connected planar domains
Analysis of PDEs
2026-04-06 v1
Abstract
We show the existence of planar domains with one hole for which the first non-trivial Neumann eigenfunction has a closed nodal line fully contained inside the domain. This is optimal, as it is known since Pleijel's 1956 result that the nodal line cannot be closed on simply-connected planar domains. A part of the proof is based on the study of convergence of eigenvalues and eigenfunctions of graph-like domains towards metric graphs. We improve the known results of convergence of eigenfunctions, by showing a strong transversal convergence.
Keywords
Cite
@article{arxiv.2604.03169,
title = {Neumann's nodal line may be closed on doubly-connected planar domains},
author = {Pedro Freitas and Roméo Leylekian},
journal= {arXiv preprint arXiv:2604.03169},
year = {2026}
}
Comments
24 pages, including 8 figures and the appendix