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A toy Neumann analogue of the nodal line conjecture

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Abstract

We introduce an analogue of Payne’s nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain \(\Omega \) with rotational symmetry of order two (i.e. \(x\in \Omega \) iff \(-x\in \Omega \)) “should normally” be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.

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Authors and Affiliations

  1. Group of Mathematical Physics, University of Lisbon, Campo Grande, Edifício C6, 1749-016, Lisbon, Portugal

    J. B. Kennedy

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  1. J. B. Kennedy
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Correspondence to J. B. Kennedy.

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The work of the author was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program “Investigador FCT”, reference IF/01461/2015, and project PTDC/MAT-CAL/4334/2014.

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Kennedy, J.B. A toy Neumann analogue of the nodal line conjecture. Arch. Math. 110, 261–271 (2018). https://doi.org/10.1007/s00013-017-1117-1

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  • DOI: https://doi.org/10.1007/s00013-017-1117-1

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