Critical points of the second Neumann eigenfunctions on the quadrangles with symmetry

TL;DR

This paper investigates symmetry properties of the second Neumann eigenfunctions on quadrilaterals with symmetry, providing results on eigenfunction symmetry, critical points, and affirming cases of the Hot Spots Conjecture.

Contribution

It offers new insights into the symmetry and critical point structure of second Neumann eigenfunctions on specific symmetric quadrilaterals, including isosceles trapezoids, parallelograms, and kites.

Findings

Eigenfunction symmetry depends on geometric parameters like base angle and height.

No non-vertex critical points in parallelograms, with specific symmetry properties.

Characterization of critical points and symmetry behavior in kite domains.

Abstract

In this paper, we focus primarily on the symmetry properties of the second Neumann eigenfunction $u$ with respect to the symmetry axis or symmetry center of the relevant domain $Q$, such as isosceles trapezoids, parallelograms, kite domains, and we provide some affirmative answers to the Hot Spots Conjecture for these domains. Our proofs combine symmetry decomposition, comparison of eigenvalues, and the continuity method. Precisely, we have the following three aspects of results. (1) when $Q$ is an isosceles trapezoid, if the base angle $\alpha\le \frac{\pi}{3}$, $u$ is antisymmetric about the symmetric axis; if the base angle $\alpha> \frac{\pi}{3}$, there exists a critical height $\hat{h}$, when height $h<\hat{h}$, $u$ is antisymmetric about the symmetric axis; when height $h>\hat{h}$, $u$ is symmetric about the symmetric axis; when height $h=\hat{h}$, the multiplicity of second Neumann eigenvalue is 2. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$. (2) When $Q$ is a parallelogram, $u$ is centrally antisymmetric about the center of $Q$ and does not have any non-vertex critical points. In particular, when $Q$ is a rhombus, $u$ is symmetric with respect to the longer diagonal and is antisymmetric with respect to the short diagonal. (3) When $Q$ is a kite $P_1P_2P_3P_4$, where $P_1$ is the origin, $P_2=(a,-h)$ lies in the four quadrant, $P_3=(1,0)$ lies on the positive $x$-axis, and $P_4=(a,h)$ is symmetric with $P_2$ about $x$-axis which lies in the first quadrant. If $a\ge 2$, $u$ is antisymmetric about $x$-axis; if $0<a<2$, there exist two constants $h_0$ and $h_1$ ($h_0\le h_1$), when $h<h_0$, $u$ is symmetric about $x$-axis; when $h>h_1$, $u$ is antisymmetric about $x$-axis. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$.