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

2025/10/28 by Freitas, Pedro, Leylekian, Roméo
#35B05 #35J25 #35P05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.24436

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.

Citations

• On the effects of small perturbation on low energy Laplace eigenfunctions
• On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains--with some numerics--
• Closed nodal surfaces for simply connected domains in higher dimensions
• Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces

Related

• A toy Neumann analogue of the nodal line conjecture
• Closed nodal surfaces for simply connected domains in higher dimensions
• Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions
• The nodal line of the second eigenfunction of the Robin Laplacian in ℝ² can be closed
• A counterexample to the "hot spots" conjecture