Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites

Abstract

In this paper, we determine the symmetry properties and non-vertex critical points of the second Neumann eigenfunction $u$, as well as the multiplicity of the corresponding eigenvalue, on isosceles trapezoids and kites. By exploiting reflection symmetry, we reduce the problem to a comparison between the second Neumann eigenvalue and the first mixed Dirichlet--Neumann eigenvalue on the half-domain. More precisely, for isosceles trapezoids with base angle $\alpha\leq \frac{\pi}{3}$, the second eigenfunction is antisymmetric. If $\frac{\pi}{3}<\alpha<\frac{\pi}{2}$, there exists a critical height $\hat{h}(\alpha)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(\alpha)$ and symmetric when $h>\hat{h}(\alpha)$, while at $h=\hat{h}(\alpha)$ the second Neumann eigenvalue has multiplicity two. For a convex kite $P_1P_2P_3P_4$, where $P_1=(0,0)$, $P_2=(a,-h)$, $P_3=(1,0)$, and $P_4=(a,h)$, an analogous result holds: there exists a critical height $\tilde{h}(a)$ such that $u$ is symmetric with respect to the $x$-axis when $h<\tilde{h}(a)$ and antisymmetric with respect to the $x$-axis when $h>\tilde{h}(a)$, while at $h=\tilde{h}(a)$ the second Neumann eigenvalue has multiplicity two. In all cases where the second Neumann eigenvalue is simple, we determine all non-vertex critical points of the corresponding eigenfunction, thereby verifying the hot spots conjecture.