I'm studying Green functions on a ball (that I call $G$). I know that this function is harmonic, and that all their partial derivatives are harmonic (a consequence of representation formula).
Can I say that the directional derivative $$\frac{\partial G}{\partial \eta}$$ is harmonic? ($\eta$ is the outward unitary vector of the ball). I think it's true, but I'm not sure.
Thanks a lot,
In fact the affirmation is true.
If $y \in \partial B(0,R)$ and $x \in $B(0,R)$, we have
$$\frac{\partial G}{\partial \eta}(x,y)= -\frac{1}{n \alpha(n)R} \frac{R^2-|x|^2}{|x-y|^n}.$$
As $-\frac{1}{n \alpha(n)R}$ is constant, it is enough to calculate the directional derivative of $$uv:= (R^2-|x|^2)|x-y|^{-n}.$$
After a lot of standard calculations, we conclude that, in fact, $\Delta (uv)=0$.