(i) Let $V$ be a bounded region in $\mathbb{R}^{3}$ with smooth boundary $S=\partial V$. Show that Poisson's equation in $V$

$$\nabla^{2} u=\rho$$

has at most one solution satisfying $u=f$ on $S$, where $\rho$ and $f$ are given functions.

Consider the alternative boundary condition $\partial u / \partial n=g$ on $S$, for some given function $g$, where $n$ is the outward pointing normal on $S$. Derive a necessary condition in terms of $\rho$ and $g$ for a solution $u$ of Poisson's equation to exist. Is such a solution unique?

(ii) Find the most general spherically symmetric function $u(r)$ satisfying

$$\nabla^{2} u=1$$

in the region $r=|\mathbf{r}| \leqslant a$ for $a>0$. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at $r=a$ : (a) $u=0$, (b) $\frac{\partial u}{\partial n}=0$.

Compare these with your results in part (i).