Computation of the logarithmic capacity

Computing the logarithmic capacity of a compact subset of the complex plane is a notoriously difficult task. We will discuss its computation for (1) sets with finitely many components and with piece-wise smooth boundary, (2) sets with many small components. In case (1), our method relies on Walsh's conformal map onto lemniscatic domains. The logarithmic capacity is one of the parameters of the lemniscatic domain and can be computed separately from the conformal map. In case (2), we obtain the logarithmic capacity from an approximation of the Green function with the charge simulation method (or method of fundamental solutions). We give several examples and compute in particular (an approximation of) the logarithmic capacity of the Cantor set.
This talk is based on joint work with Joerg Liesen (TU Berlin) and Mohamed M.S. Nasser (Qatar University).