Uniqueness of an electrostatic dielectric boundary-value problem

For positive scalar permittivities and specified free charges, fixed conductor potentials, dielectric matching conditions and decay at infinity, the electrostatic field is unique. Subtract two proposed potentials and integrate $\nabla \cdot (ϵ\delta \varphi \nabla \delta \varphi )$. The boundary terms vanish or cancel across dielectric interfaces, leaving $\int ϵ|\nabla \delta \varphi {|}^{2}dV=0$. Hence the difference field is zero, with any potential constant fixed by the boundary convention.

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.