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

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. Mathematics subject classification (2010): Primary 35P15. Secondary 35J20.