On continuity of logarithmic capacity

The studies of the continuity of set capacity and related quantities have a long history in potential theory. In 1961, Gehring proved that the conformal modulus of planar annuli is continuous under Hausdorff convergence of the boundary components. Aseev proved the continuity of condenser capacity under the Hausdorff convergence of its plates, under the assumption that the plates are uniformly perfect with the same constant. Later, Aseev and Lazareva proved the analogous continuity result for logarithmic capacity of sets. Ransford, Younsi, and Ai recently proved that the logarithmic capacity of a set varies continuously under holomorphic motions.
In the talk, we consider questions about the convergence of Green's functions under the convergence of domains in the sense of the kernel and about the continuity of logarithmic capacity when the Hausdorff distance to the limit set tends to zero at a sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition.
This talk is based on joint work with L. V. Kovalev.