 Solution

Suppose ${\varphi }_1,{\varphi }_2$ are two sufficiently regular solutions with the same Dirichlet boundary condition, and put $w={\varphi }_1-{\varphi }_2$. Then $\Delta w=0$ in $V$ and $w=0$ on its boundary. Apply the divergence theorem to $w\nabla w$: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 uniqueness of the Dirichlet solution, without assuming its existence.

For a prescribed Neumann boundary condition, the difference instead has ${\partial }_{n}w=0$. The same energy argument still gives $\nabla w=0$, but does not fix the constant. Consequently 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 $g={\partial }_n\varphi$ on $S$, the divergence theorem givesIt 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.