 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.