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,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 a technique that simulates a Schrödinger potential.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.
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.
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