Payne's nodal line conjecture fails on doubly-connected planar domains
Analysis of PDEs
2025-10-29 v1
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.
Keywords
Cite
@article{arxiv.2510.24436,
title = {Payne's nodal line conjecture fails on doubly-connected planar domains},
author = {Pedro Freitas and Roméo Leylekian},
journal= {arXiv preprint arXiv:2510.24436},
year = {2025}
}
Comments
13 pages, including 5 figures and the appendix