Given that $ \Omega \in \mathbb{R}^n $, $ \partial\Omega $ is a closed hypersurface, and $ u \in C^2(\Omega) \cap C^1(\overline{\Omega}) $ satisfies
$$ \large\begin{cases} \Delta u = 0, \>\> x \in \Omega \\ \frac{\partial u}{\partial\nu}\Bigg|_{\partial\Omega} = \psi(x), \>\>x \in \partial\Omega \end{cases} $$
I have to prove that there are at least two points $ x_1, x_2 \in \partial\Omega $, such that $ \psi(x_1) = 0 $ and $ \psi(x_2) = 0 $. I assume that $ \nu $ is an outer normal unit vector to $ \partial\Omega $. My idea was to apply the Green's formula, but I'm not sure how to proceed further.
Any pointers would be highly appreciated!