Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 3 / 12C / i

For two solutions of the Poisson equation with the same Dirichlet boundary condition, let $w=u_1-u_2$. Then ${\nabla }^{2}w=0$ in $V$ and $w=0$ on its boundary. By Green's first identity,Thus $w$ is constant on each connected component, and the zero boundary values make each constant zero. The Dirichlet solution is unique whenever it exists. This argument requires the regularity needed for the displayed integration by parts; it does not assert existence for arbitrary rough data.

With a Neumann boundary condition, integrating the Poisson equation and using the divergence theorem instead gives the necessary compatibility conditionOn a disconnected region this condition must hold on each connected component separately. If two solutions have the same normal derivative, their difference is harmonic with ${\partial }_{n}w=0$. The same Green's first identity again forces $\nabla w=0$. Thus Neumann solutions are unique only up to an additive constant on each connected component. Adding a constant preserves both the equation and normal derivative, so uniqueness without a normalization cannot hold. On a connected region, prescribing the mean value removes this one-dimensional freedom.