### Abstract

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n > 3. The result fails for n = 2, because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle.

Original language | English (US) |
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Pages (from-to) | 381-394 |

Number of pages | 14 |

Journal | Mathematical Inequalities and Applications |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2012 |

### Keywords

- Free membrane
- Isodiametric
- Isoperimetric

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Laugesen, R. S., Pan, Z. C., & Son, S. S. (2012). Neumann eigenvalue sums on triangles are (Mostly) minimal for equilaterals.

*Mathematical Inequalities and Applications*,*15*(2), 381-394. https://doi.org/10.7153/mia-15-32