There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky:
Suppose some volume $\Omega$ is not empty but contains several charged bodies, dielectrics or conductors. For each conducting body, we specify either the net electric charge or the potential (which is constant over a conductor). For each dielectric body, we specify the entire charge distribution, i.e. $\rho(x,y,z)$ as a function of position within the body. Finally, we specify the potential $V (x,y,z)$ along the outer boundary $S$ of the volume $\Omega$. Under these conditions, there is a unique solution for the potential $V(x,y,z)$ inside $\Omega$ — including the dielectric and the conducting bodies themselves.
The theorem and its proof can be found here.
Notice that the important thing here is that only the NET charges on the conductors are specified, not their charge distributions (otherwise this would be nothing more than the uniqueness of the solution to the Poisson equation with mixed Dirichlet and Neumann boundary conditions). To be more precise, the real difficulty is to show that specifying the net charge on the conductors amounts to specifying the charge distribution on their surfaces.
This theorem is by no mean trivial, as seem to believe many authors that content themselves to invoke the "linearity and homogeneity of the equations of electrostatics" in many situations, e.g. when establishing the existence of the capacitance matrix.
I have no idea if Dr Kaplunovsky demonstrated this theorem by himself, or if, more likely, he found it somewhere.
Are you aware of any book or reliable source where it can be found, in order to be sourced?
