Neumann's nodal line may be closed on 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.

Figures (8)

• Figure 1: Schematic representation of a domain with a second Neumann eigenfunction whose nodal line (in blue) is expected to remain closed.
• Figure 2: Illustration of the Dirichlet FDSH constructed in freitas-leylekian. In each case, the dotted line is the segment bisecting the rectangle and the blue line is the expected nodal line.
• Figure 3: Two graphs described in the introduction.
• Figure 4: A planar caterpillar tree with a loop.
• Figure 5: Schematic representation of the Neumann FDSH described in the introduction. The expected nodal line is drawn in blue.

Theorems & Definitions (16)

• Conjecture : Topological Nodal Line Conjecture
• proof
• Proposition 2.3
• proof : Proof of Theorem \ref{['thm:resultat principal']}
• proof
• proof
• Proposition 3.5: Localisation of the nodal set
• proof
• proof
• Proposition 5.1