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

2011/02/01 by Richard S. Laugesen, Laugesen, R. S., Zehan Pan +3
Materials Science · Mathematics · #35P15 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1102.0071

openalex publication_date 2011/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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