Far from the rim of the inner disk one has a good regularity. At the rim though, I think you can get the solution behaviour from explicit constructions of potentials induced by uniformly charged ellipsoids and collapsing one of axes, e.g., as given in Kellogg's book.

Far from the rim of the inner disk one has a good regularity. At the rim though, I think you can get the solution behaviour from explicit constructions of potentials induced by a charged ellipsoid and collapsing one of axes, e.g., as given in Kellogg's book.