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

TL;DR

This paper demonstrates that on certain doubly-connected planar domains, the first non-trivial Neumann eigenfunction can have a closed nodal line entirely inside the domain, extending previous results.

Contribution

It constructs examples of doubly-connected domains with closed internal nodal lines for the first non-trivial Neumann eigenfunction, and advances convergence analysis of eigenfunctions on graph-like domains.

Findings

Existence of domains with closed internal nodal lines for Neumann eigenfunctions.

Improved understanding of eigenfunction convergence on graph-like domains.

Extension of classical results to doubly-connected planar domains.

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.