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arXiv · 2103.02276

A counterexample to Payne's nodal line conjecture with few holes

Joel Dahne·Javier Gómez-Serrano·Kimberly Hou
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Abstract

Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with many holes and raised the question of the minimum number of holes a counterexample can have. In this paper we prove it is at most 6.

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Joel Dahne, Javier Gómez-Serrano, Kimberly Hou. 2021-03-03. A counterexample to Payne's nodal line conjecture with few holes. https://doi.org/10.1016/j.cnsns.2021.105957

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