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

TL;DR

This paper demonstrates that Payne's nodal line conjecture, which predicts the behavior of nodal lines of eigenfunctions, does not hold for doubly-connected planar domains, only for simply-connected ones.

Contribution

The authors provide counterexamples of doubly-connected domains where the second Dirichlet eigenfunction's nodal line is closed and does not touch the boundary, disproving the conjecture's generality.

Findings

Counterexamples in doubly-connected domains

Nodal lines can be closed and interior in such domains

Conjecture holds only for simply-connected domains

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.