Are these the only first eigenfunctions on a hemisphere?

Question

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$, and the associated eigenfunctions are the coordinate functions $x,y,z$ restricted to the sphere. I wonder if the functions $x$ and $y$ generate the first eigenspace of the Laplacian the upper hemisphere with Neumann boundary condition. It is easy to see that $z$ does not satisfy Neumann boundary condition.

Answer 1

As Christian Remling already indicated in the comments: one can use reflection techniques and thus show: eigenfunctions of the Laplace-Beltrami operator on the $n$-dimensional hemisphere with Neumann boundary conditions are in 1-to-1 correspondence to eigenfunctions on the $n$-dimensional sphere that are invariant under reflection $(x_1,x_2,..,x_n,z)\mapsto (x_1,x_2,...,x_n,-z)$. The isomporphism is given by symmetric extension, thus its inverse is restriction. As the eigenfunctions on the sphere are explicitly known as restrictions of harmonic homogeneous polynomials on $\mathbb{R}^{n+1}$, one knows the eigenfunctions explicitly [see e.g. M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variété Riemannienne, Lecture Notes in Mathematics 194, Springer Verlag, Berlin, New York, 1971]. E.g. the first eigenspace is generated by the constant function $1$, the second one by the restrictions of the $x_i$. This holds in any dimension $n\geq 1$.