Uniqueness theorem for Poisson's equation

Remarks

The difference $w = \varphi_1 - \varphi_2$ is harmonic in $\Omega$. By the first of Green's identities, $\int_\Omega \abs{\nabla w}^2 \, dx = \int_{\partial \Omega} w \, \partial_n w \, dA$, and the right side is $0$ under either boundary condition. So $\nabla w = 0$, and $w$ is constant because $\Omega$ is connected; in case 1 the constant is $0$ since $w = 0$ on $\partial \Omega$. The same argument gives uniqueness for mixed data, with $\varphi$ given on one part of the boundary and $\partial_n \varphi$ on the rest. In case 2 the constant cannot be removed: $\varphi$ and $\varphi + 1$ have the same normal derivative. On an unbounded region, uniqueness needs a condition at infinity, such as $\varphi(x) \to 0$ as $\abs{x} \to \infty$.

Going deeper

For a student

The theorem gives uniqueness for the Dirichlet problem and, up to constants, for the Neumann problem of Poisson's equation. In electrostatics it justifies guessing: the method of image charges and separation of variables work because any solution found is the solution. It also shows that the field inside a closed hollow conductor without charge is zero. Existence is a separate question, answered by potential theory or by variational methods.