### Abstract

We extend the two-dimensional results of Jerison (2000) on the location of the nodal set of the first Neumann eigenfunction of a convex domain to higher dimensions. If a convex domain Ω in ℝ^{n} is contained in a long and thin cylinder [0, N] × B_{∈}(0) with nonempty intersections with {x_{1} = 0} and {x_{1} = N}, then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to ∈, whose coefficients are expressed in terms of the volume of the cross sections of the domain. Also, the first nodal set is located within a distance comparable to ∈ near the zero of the corresponding ordinary differential equation.

Original language | English (US) |
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Pages (from-to) | 5111-5137 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2009 |

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### Keywords

- Convex domains
- Eigenfunctions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*361*(10), 5111-5137. https://doi.org/10.1090/S0002-9947-09-04729-1