Nodal line conjecture

In mathematics, the nodal line conjecture is a statement posed in 1967 by Lawrence E. Payne about the Laplacian partial differential equation. The original conjecture predicts that for the Dirichlet problem on a bounded two-dimensional domain, the second eigenfunction has a nodal line that meets the boundary of the domain. The general conjecture was proved false in 1997 by carving a large number of microscopic holes out of a disk,^[1] a technique that simulates a Schrödinger potential.^[2]^[3]

Other positive and negative results are known for various special cases of domains; in general, it remains an open problem to describe how simple the domain must be for the conjecture to hold.^[4]

References

^ M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili (December 15, 1997) "The nodal line of the second eigenfunction of the Laplacian in R^2 can be closed" Duke Math. J. 90(3): 631-640. DOI: 10.1215/S0012-7094-97-09017-7
^ D. Cioranescu and F. Murat. "Un terme étrange venu d’ailleurs", I and II. In Nonlinear Differential Equations and Their Applications, Collège de France Seminar, Vol. 60 and 70, Research Notes in Mathematics, pages 98–138, 154–178. Pitman, London, 1982–1983.
^ Simons Foundation (2026-03-13). Jaume de Dios Pont — Some Extreme Regimes of the Laplace Operator. Retrieved 2026-03-20 – via YouTube.
^ Dahne, Joel; Gómez-Serrano, Javier; Hou, Kimberly (2021-12-01). "A counterexample to Payne's nodal line conjecture with few holes". Communications in Nonlinear Science and Numerical Simulation. 103: 105957. doi:10.1016/j.cnsns.2021.105957. ISSN 1007-5704.{{cite journal}}: CS1 maint: article number as page number (link)

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[1] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili (December 15, 1997) "The nodal line of the second eigenfunction of the Laplacian in R^2 can be closed" Duke Math. J. 90(3): 631-640. DOI: 10.1215/S0012-7094-97-09017-7

[2] D. Cioranescu and F. Murat. "Un terme étrange venu d’ailleurs", I and II. In Nonlinear Differential Equations and Their Applications, Collège de France Seminar, Vol. 60 and 70, Research Notes in Mathematics, pages 98–138, 154–178. Pitman, London, 1982–1983.

[3] Simons Foundation (2026-03-13). Jaume de Dios Pont — Some Extreme Regimes of the Laplace Operator. Retrieved 2026-03-20 – via YouTube.

[4] Dahne, Joel; Gómez-Serrano, Javier; Hou, Kimberly (2021-12-01). "A counterexample to Payne's nodal line conjecture with few holes". Communications in Nonlinear Science and Numerical Simulation. 103: 105957. doi:10.1016/j.cnsns.2021.105957. ISSN 1007-5704.{{cite journal}}: CS1 maint: article number as page number (link)

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