The nodal line conjecture formulated by L. Payne in 1967 states that the nodal line of any second eigenfunction of the
Dirichlet Laplacian on bounded planar domains must touch the boundary of the domain at two (distinct) oints.
A first counterexample was given by M. and T. Hoffmann-Ostenhof and N. Nadirashvili in 1997. However, their example
contained an unspecified number of holes, and they raised the question of what the smallest number of holes in a
counterexaple could be. In a recent paper [FL25], we have shown that it is, in fact, sufficient to have a single hole
to produce a counterexample.
Following an argument used by Pleijel in a 1956 paper, it had been thought for a long time that such counterexamples
could not exist for Neumann boundary conditions. However, based on an idea similar to that in the Dirichlet case, albeit
using quite different techniques, we have now also showed that there exist doubly-connected planar domains for which
the eigenfunction associated with the first nontrivial Neumann eigenvalue has a closed nodal line which does not touch
the boundary.
In [F02] we showed that both the nodal line and the hot spots conjectures fail on simply-connected surfaces.
[F02] Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces. Indiana Univ. Math. J. 51 (2002), 305–316.
[FK07] Unbounded planar domains whose second nodal line does not touch the boundary. Math. Res. Lett. 14 (2007), 107–111 (with David Krejčiřík).
[FK08] Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57 (2008), 343–376 (with David Krejčiřík).
[FL25] Payne’s nodal line conjecture fails on doubly-connected planar domains. Preprint at https://arxiv.org/abs/2510.24436. 2025. (with Roméo Leylekian).
[FL26] Neumann’s nodal line may be closed on doubly-connected planar domains. (with Roméo Leylekian).