= Solution
Apply <Green second identity> to $u$ and the free-space <Green function> $G_{fs}$ on $V$ with a small ball about $\mathbf r_0$ removed. Both functions are harmonic there, so only boundary terms remain. The contribution from the small sphere tends to $u(\mathbf r_0)$ because
$$
G_{fs}(\mathbf r;\mathbf r_0)=-\frac1{4\pi|\mathbf r-\mathbf r_0|}
$$
has unit delta source, while the remaining small-sphere term vanishes. Taking the radius to zero gives <Green's third identity>
$$
\boxed{
u(\mathbf r_0)=
\int_S\left(u\frac{\partial G_{fs}}{\partial n}
-G_{fs}\frac{\partial u}{\partial n}\right)dS}.
$$
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