Euclidean Triangles Have No Hot Spots

TL;DR

This paper proves that for any Euclidean triangle, the second Neumann eigenfunction does not have critical points inside the triangle, revealing a fundamental property of eigenfunctions in geometric domains.

Contribution

It establishes a novel result that second Neumann eigenfunctions in Euclidean triangles lack interior critical points, advancing understanding of eigenfunction behavior in polygons.

Findings

Second Neumann eigenfunction has no interior critical points in Euclidean triangles.

Provides insight into the structure of eigenfunctions in polygonal domains.

Enhances theoretical understanding of spectral properties of triangles.

Abstract

We show that a second Neumann eigenfunction u of a Euclidean triangle has no critical points in the interior of the triangle.