Logarithmic capacity of chains of circles

We study extremal problems on the logarithmic capacity of arrangements of n ≥ 2 n\ge 2 disks in the complex plane. We solve some of them, including the problem on the minimal logarithmic capacity over the sets of n ≥ 3 n\ge 3 disks D k D_k covering the unit circle T \mathbb {T} and such that their boundary circles ∂ D k \partial D_k are orthogonal to T \mathbb {T} , and problems on the maximal logarithmic capacity over the set of all strings consisting of n n disks each of radius 1 1 and over the set of all necklaces of n n disks of radius 1 1 . Several other problems remain open and the most difficult of them is the problem on the minimal logarithmic capacity over the sets of n n disks with disjoint interiors of radius 1 1 each. We demonstrate a few methods, which can be applied to study these problems, and explain limitations of these methods and possible steps toward solutions of some of the remaining open problems.