### Abstract

We consider the zeros on the boundary ∂Ω of a Neumann eigen- function φ_{λj} of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(Ω_{j}) where −Δφ_{λj} = λ^{2}_{j}φ_{λj}. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(Ω_{j}). It follows that the number of nodal lines of φ_{λj} (components of the nodal set) which touch the boundary is of order Ω_{j} . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

Original language | English (US) |
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Pages (from-to) | 649-686 |

Number of pages | 38 |

Journal | Journal of Differential Geometry |

Volume | 81 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2009 |

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Journal of Differential Geometry*,

*81*(3), 649-686. https://doi.org/10.4310/jdg/1236604347