Maximizing Neumann fundamental tones of triangles

TL;DR

This paper establishes sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangles, showing the equilateral triangle maximizes the first nonzero eigenvalue among all triangles with fixed perimeter or area.

Contribution

It proves that the equilateral triangle maximizes the first nonzero Neumann eigenvalue and related means among all triangles with fixed perimeter or area, providing new extremal inequalities.

Findings

The first nonzero Neumann eigenvalue is maximized by the equilateral triangle for fixed perimeter.

Similar maximization results hold for the harmonic and arithmetic means of the first two eigenvalues.

The results are sharp and extend classical isoperimetric inequalities to eigenvalues of triangles.

Abstract

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.