Uniqueness of equilibria in electrostatics I have the following problem.
Suppose we place some continuous charge distribution (with total charge $Q$)on some conducting domain. The charge will redistribute itself on the domain until it's in equilibrium.
Is the eventual equilibrium independent of the charge distribution? In other words, does the equilibrium depend on only the total charge $Q$ and the shape of the domain?
Aside questions: Does it matter if the domain is 2d or 3d? Does convexity matter?
Note that in the case of point charges, there can be more than one unique solution. See: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=BEFB88179758A936195EB3C763C6D64E?doi=10.1.1.5.9566&rep=rep1&type=pdf
Paper: Minimum energy point charge configurations on a circular disk, from the Helsinki University of Technology.
 A: For any given conductor, if we dump a small "lump" of charge on it and allow it to disperse, then there will only be one possible outcome. This follows from the uniqueness theorem for conductors. When the charge finds it's equilibrium, we know (from the properties of conductors) that the surface will be at a constant potential. Hence, there will be a unique electric field and so there must be a unique charge distribution, regardless of how we dump that charge on the conductor.
Then what's the problem for the case of point charges? It's a subtle matter; it's because we can't be dealing with a conductor in this case. Note that if we have a neutral "conductor" and we put point charges on it, then allow the point charges to distribute themselves, clearly the potential is not constant on the surface! To see this, imagine a disk shaped conductor with two charges at opposite ends of the boundary. Clearly in this case the potential is not constant everywhere. This holds similarly for all conductors. In short, we can't apply our uniqueness laws.
This makes sense on physical grounds; a conductor is defined as a substance with an "infinite" number of point charges. So if we put 1 point charge on a conductor, it should not be distinguishable from all the other point charges. There are so many point charges on a conductor that the charge should be thought of as continuous. 
So in the paper linked in the question above, the disk cannot be a conductor. It should instead be thought of as a certain portion of free space with a specified boundary.
