The Dirichlet eigenvalue problem on a given suitable domain $\Omega\subset\mathbb{R}^n$ asks one to find such function(s) $u$ and eigenvalue(s) $\lambda$ that
$$\begin{cases}\Delta u &= \lambda u\text{, in interior of $\Omega$}\\\ u &= 0\text{, on $\partial \Omega$}\end{cases}$$
with $\Delta$ being an appropriate version of the Laplace-Beltrami operator on $\Omega$. In the case of the unit sphere $S^2$, it has been shown that the only eigenfunctions of spherical Laplacian, given in the angular coordinates $(\theta, \phi)\in \left(0, 2\pi\right)\times \left(0, \pi\right)$ as
$$\Delta f = (\sin(\theta))^{-1}\left(\left(\sin(\theta)f_\theta\right)_\theta + \left(\sin(\theta)\right)^{-1}f_{\phi\phi}\right)$$
are the functions
$$f_{m,n}\left(\theta, \phi\right) = \exp(im\phi)P_n^m\left(\cos(\theta)\right), n \in\mathbb{N}, -n\leq m\leq n$$
with
$$P_n^m(x) = i^m\frac{\Gamma(n + m + 1)}{\pi\Gamma(n + 1)}\int_0^\pi\left(x + \sqrt{x^2 - 1}\cos(\phi)\right)^n\cos(m\phi)d\phi$$
Well, as the $\exp$ function never vanishes, $f_{m,n}(\theta, \phi) = 0$ if and only if $P_n^m\left(\cos(\theta)\right) = 0$. But then $f_{m,n}$s can only vanish on some points of the boundary of $\left(0, 2\pi\right)\times \left(0, \pi\right)$ and not for all. So does the Dirichlet eigenvalue problem make sense on the sphere $S^2$? I am not asking this as an attempt to show any "paradoxy", but rather as an honest attempt to understand whether one can faithfully talk about the original Dirichlet eigenfunction problem on a manifold such as the sphere $S^2$.