Critical points of Laplace eigenfunctions on polygons

TL;DR

This paper investigates the critical points of Laplace eigenfunctions on polygons, revealing conditions under which interior critical points exist or are constrained to vertices, especially for the second Neumann eigenfunction.

Contribution

It establishes a link between the absence of interior critical points in convex quadrilaterals and the existence of unstable critical points, and characterizes critical points on Lip-1 polygons.

Findings

Convex quadrilaterals with no interior critical points have an associated unstable critical point.

Critical points on Lip-1 polygons with no orthogonal sides are located at acute vertices.

The study provides conditions for the existence and location of critical points of Laplace eigenfunctions.

Abstract

We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.