(i) Define the concept of "fundamental solution" of a linear constant-coefficient partial differential operator and write down the fundamental solution for the operator $-\Delta$ on $\mathbb{R}^{3}$.

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

(iii) Let $u \in C^{2}\left(\mathbb{R}^{3}\right)$ be a harmonic function which satisfies $u(p) \geqslant 0$ at every point $p$ in an open set $\Omega \subset \mathbb{R}^{3}$. Show that if $B(z, r) \subset B(w, R) \subset \Omega$, then

$$u(w) \geqslant\left(\frac{r}{R}\right)^{3} u(z) .$$

Assume that $B(x, 4 r) \subset \Omega$. Deduce, by choosing $R=3 r$ and $w, z$ appropriately, that

$$\inf _{B(x, r)} u \geqslant 3^{-3} \sup _{B(x, r)} u .$$

[In (iii), $B(z, \rho)=\left\{x \in \mathbb{R}^{3}:\|x-z\|<\rho\right\}$ is the ball of radius $\rho>0$ centred at $\left.z \in \mathbb{R}^{3} \cdot\right]$