Uniqueness theorem for the electrostatic potential

In a vacuum domain with specified charge density, fixed boundary electric potential and the appropriate condition at infinity, two solutions of the Poisson equation have difference $w$ satisfying ${\nabla }^{2}w=0$ with zero boundary data. Integration by parts gives $\int |\nabla w{|}^{2}dV=0$; hence their electric fields coincide, and their potentials coincide after fixing the constant. This is the Dirichlet case of the more general dielectric uniqueness result. It justifies the exterior field constructed by the method of images.