= Solution
Suppose $\phi_1,\phi_2$ are two sufficiently regular solutions with the same <Dirichlet boundary condition>, and put $w=\phi_1-\phi_2$. Then $\Delta w=0$ in $V$ and $w=0$ on its boundary. Apply the <divergence theorem> to $w\nabla w$:
$$
\int_V\bigl(|\nabla w|^2+w\Delta w\bigr)\,dV=\int_S w\,\partial_nw\,dS=0.
$$
Thus $\nabla w=0$ throughout $V$, so $w$ is constant on each connected component. Its zero boundary values force every such constant to be zero. This proves \b[uniqueness of the Dirichlet solution], without assuming its existence.
For a prescribed <Neumann boundary condition>, the difference instead has $\partial_nw=0$. The same energy argument still gives $\nabla w=0$, but does not fix the constant. Consequently \b[the solution is unique up to an additive constant on each connected component]. A value at one point, or the integral mean, fixes the constant on a connected region. There is also a necessary compatibility condition: if $g=\partial_n\phi$ on $S$, the <divergence theorem> gives
$$
\boxed{\int_S g\,dS=\int_V\Delta\phi\,dV=0}.
$$
It is required on each component if the region is disconnected. Data failing this condition admit no harmonic solution. This is the homogeneous-source instance of the <Neumann Poisson problem>.
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