Boundary contact of the second nodal set in convex domains
Let $d \ge 3$ and let $\Omega \subset \mathbb{R}^d$ be a bounded convex domain with $C^\infty$ boundary. Let $0 < \lambda_1 < \lambda_2 \le \lambda_3 \le \cdots$ be the Dirichlet eigenvalues of $-\Delta$ on $\Omega$, counted with multiplicity. A second eigenfunction is a real nonzero $u$ with $-\Delta u = \lambda_2 u$ in $\Omega$ and $u = 0$ on $\partial\Omega$; its nodal set $\{x \in \Omega : u(x) = 0\}$ consists of interior zeros only, not of the boundary zeros imposed by the Dirichlet condition. Payne conjectured that in the plane the nodal line of a second eigenfunction cannot be a closed curve inside the domain. For convex planar domains this was proved by Melas, for $C^\infty$ boundary, and by Alessandrini in general. Damascelli proved in every dimension that the nodal set meets the boundary for domains that are symmetric and convex with respect to one direction, extending a planar theorem of Payne. Without convexity the property fails: Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili found multiply connected planar domains, Fournais domains in higher dimensions, and Kennedy, for every $d \ge 3$, a domain homeomorphic to a ball, each with a second eigenfunction whose nodal set has closure disjoint from the boundary.
Problem. Is $$\overline{\{x \in \Omega : u(x) = 0\}} \cap \partial\Omega \ne \varnothing,$$ the closure taken in $\mathbb{R}^d$, for every $d \ge 3$, every such $\Omega$ and every second eigenfunction $u$?
Source: Payne (1967), Conjecture 5, for planar domains; asked in higher dimensions by Yau (1993), Problem 45. Stated by AIM — Open Applied Problems for smooth convex domains in dimension d ≥ 3, after Freitas and Leylekian (2025).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, convex domains, Dirichlet eigenfunctions, nodal sets
References
@article{payne1967,
author = {Payne, L. E.},
title = {Isoperimetric inequalities and their applications},
journal = {SIAM Rev.},
volume = {9},
number = {3},
year = {1967},
pages = {453--488},
doi = {10.1137/1009070},
}
@incollection{yau1993,
author = {Yau, Shing-Tung},
title = {Open problems in geometry},
booktitle = {Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990)},
series = {Proc. Sympos. Pure Math.},
volume = {54, Part 1},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1993},
pages = {1--28},
doi = {10.1090/pspum/054.1/1216573},
}
@misc{freitas2025,
author = {Freitas, Pedro and Leylekian, Rom{\'e}o},
title = {Payne's nodal line conjecture fails on doubly-connected planar domains},
year = {2025},
eprint = {2510.24436},
archivePrefix = {arXiv},
}
@article{payne1973,
author = {Payne, Lawrence E.},
title = {On two conjectures in the fixed membrane eigenvalue problem},
journal = {Z. Angew. Math. Phys.},
volume = {24},
number = {5},
year = {1973},
pages = {721--729},
doi = {10.1007/BF01597076},
}
@article{melas1992,
author = {Melas, Antonios D.},
title = {On the nodal line of the second eigenfunction of the {L}aplacian in $\mathbf{R}^2$},
journal = {J. Differential Geom.},
volume = {35},
number = {1},
year = {1992},
pages = {255--263},
doi = {10.4310/jdg/1214447811},
}
@article{alessandrini1994,
author = {Alessandrini, Giovanni},
title = {Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains},
journal = {Comment. Math. Helv.},
volume = {69},
number = {1},
year = {1994},
pages = {142--154},
doi = {10.1007/BF02564478},
}
@article{damascelli2000,
author = {Damascelli, Lucio},
title = {On the nodal set of the second eigenfunction of the {L}aplacian in symmetric domains in $\mathbf{R}^N$},
journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.},
volume = {11},
number = {3},
year = {2000},
pages = {175--181},
}
@article{hoffmannostenhof1997,
author = {Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T. and Nadirashvili, N.},
title = {The nodal line of the second eigenfunction of the {L}aplacian in $\mathbb{R}^2$ can be closed},
journal = {Duke Math. J.},
volume = {90},
number = {3},
year = {1997},
pages = {631--640},
doi = {10.1215/S0012-7094-97-09017-7},
}
@article{fournais2001,
author = {Fournais, S{\o}ren},
title = {The nodal surface of the second eigenfunction of the {L}aplacian in $\mathbf{R}^D$ can be closed},
journal = {J. Differential Equations},
volume = {173},
number = {1},
year = {2001},
pages = {145--159},
doi = {10.1006/jdeq.2000.3868},
}
@article{kennedy2013,
author = {Kennedy, James},
title = {Closed nodal surfaces for simply connected domains in higher dimensions},
journal = {Indiana Univ. Math. J.},
volume = {62},
number = {3},
year = {2013},
pages = {785--798},
doi = {10.1512/iumj.2013.62.4975},
eprint = {1009.1502},
archivePrefix = {arXiv},
}
@misc{aim-convex-higher-dimensional-payne,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Boundary contact of the second nodal set in convex domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/117-convex-higher-dimensional-payne.md},
note = {Entry 117 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00336, "Boundary contact of the second nodal set in convex domains", clopen.solutions.