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arXiv · 2604.03169

Neumann's nodal line may be closed on doubly-connected planar domains

Pedro Freitas·Roméo Leylekian
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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.

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Pedro Freitas, Roméo Leylekian. 2026-04-03. Neumann's nodal line may be closed on doubly-connected planar domains. https://arxiv.org/abs/2604.03169

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