Suppose $V$ is a region in $\mathbb{R}^{3}$, bounded by a piecewise smooth closed surface $S$, and $\phi(\mathbf{x})$ is a scalar field satisfying

$$\begin{aligned} \nabla^{2} \phi &=0 \text { in } V, \\ \text { and } \phi &=f(\mathbf{x}) \text { on } S . \end{aligned}$$

Prove that $\phi$ is determined uniquely in $V$.

How does the situation change if the normal derivative of $\phi$ rather than $\phi$ itself is specified on $S$ ?