Consider a well-known problem of the electric field generated by a system composed of a point charge in proximity of a large earthed conductor. It is said that the potential due to an image charge satifies all the boundary conditions - it this case, constant potential on the surface of a conductor - and therefore, by the uniqueness theorem, it is the only possible potential distribution.
However, we can of course imagine other non-trivial fields satisfying the boundary condition of constant potential on the surface, for example generated by three imaginary charges such that the whole system is again symmetric.
So my question is: what is the full set of boundary conditions needed to find the unique potential distribution?
This specific example is just an illustration of a more general question: how to use Laplace/Poisson equations to describe the field of point charges? The divergence of electric field has a singularity at the point charge. Another incarnation of this problem occurs when we want to use the formula for the energy stored in the electric field - we find the energy of the electric field generated by a point charge to be infinite. Does this mean that the concept of a point charge is too idealised?