Nodal line conjecture
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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. doi:10.1016/j.cnsns.2021.105957. ISSN 1007-5704. https://www.sciencedirect.com/science/article/pii/S1007570421002690.
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