State and prove the mean value property for harmonic functions on $\mathbb{R}^{3}$.

Obtain a generalization of the mean value property for sub-harmonic functions on $\mathbb{R}^{3}$, i.e. $C^{2}$ functions for which

$$-\Delta u(x) \leqslant 0$$

for all $x \in \mathbb{R}^{3}$.

Let $\phi \in C^{2}\left(\mathbb{R}^{3} ; \mathbb{C}\right)$ solve the equation

$$-\Delta \phi+i V(x) \phi=0$$

where $V$ is a real-valued continuous function. By considering the function $w(x)=|\phi(x)|^{2}$ show that, on any ball $B(y, R)=\{x:\|x-y\|<R\} \subset \mathbb{R}^{3}$,

$$\sup _{x \in B(y, R)}|\phi(x)| \leqslant \sup _{\|x-y\|=R}|\phi(x)| .$$