Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality

TL;DR

This paper identifies the triangle shape that minimizes the first nonzero Neumann Laplacian eigenvalue among all triangles with a fixed diameter, leading to an optimal Poincaré inequality for triangles.

Contribution

It establishes the minimal eigenvalue for a specific triangle shape and derives an optimal Poincaré inequality, using symmetry and antisymmetry properties of eigenmodes.

Findings

The degenerate acute isosceles triangle minimizes the eigenvalue.

Symmetry of the fundamental mode is proven for certain angles.

Antisymmetry of the mode is shown for larger angles.

Abstract

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincar\'{e} inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than $\pi/3$. Antisymmetry is proved for apertures greater than $\pi/3$.