Neumann eigenvalue sums on triangles are (Mostly) minimal for equilaterals

R. S. Laugesen, Z. C. Pan, S. S. Son

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)381-394
Number of pages14
JournalMathematical Inequalities and Applications
Issue number2
StatePublished - Apr 2012


  • Free membrane
  • Isodiametric
  • Isoperimetric

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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