= Solution
Suppose $\phi_1,\phi_2$ solve the Dirichlet problem and put $u=\phi_1-\phi_2$. Then $\nabla^2u=0$ in $V$ and $u=0$ on $S$. The <divergence theorem> applied to $u\nabla u$ gives Green's identity
$$
\int_V|\nabla u|^2\,dV
=\int_Su\,\partial_nu\,dS-
\int_Vu\nabla^2u\,dV=0.
$$
Thus $\nabla u=0$, so $u$ is constant; its zero boundary value makes it zero. The Dirichlet solution is therefore unique.
For homogeneous Neumann data the same calculation again shows that $u$ is constant, but the <boundary condition> does not determine that constant. Hence, whenever a Neumann solution exists, adding any constant produces another solution. This is the standard <uniqueness of Poisson equation> distinction.
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