Dirichlet eigenvalue sums on triangles are minimal for equilaterals

2010/08/07 by Laugesen, Richard, Siudeja, Bartlomiej
#FOS: Mathematics #Primary 35P15. Secondary 35J20 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1008.1316

Abstract

Among all triangles of given diameter, the equilateral triangle is shown to minimize the sum of the first n eigenvalues of the Dirichlet Laplacian, for each n ≥ 1. In addition, the first, second and third eigenvalues are each proved to be minimal for the equilateral triangle. The disk is conjectured to be the minimizer among general domains.

Related

• Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals
• The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles
• Sharp bounds for eigenvalues of triangles
• Minimal Partitions for p-norms of Eigenvalues
• Isoperimetric inequalities for eigenvalues of triangles