= 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 $\nabla^2w=0$ with zero boundary data. Integration by parts gives $\int|\nabla w|^2dV=0$; 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>.
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