Capacity of a compact set in a logarithmic potential field
Let \(E\) be the union of a finite number of continua (possibly, some of them degenerate to a point), \(\mu\) a positive Borel unit measure on \(E\), and \(V^\mu\) its (spherically normalized) logarithmic potential. Let \(K\) be a compact set in \(\mathbb C\), disjoint from \(E\). The author finds a formula for the weighted capacity of \(K\) in the external field \(\psi = -V^\mu\), in terms of the Green function of the complement of \(K\) in \(\overline{\mathbb C}\). This extends his previous work, where \(E\) was a finite union of points. As an application, the following example is constructed: there exist \(a,b\notin E=[-1,1]\) and a positive Borel unit measure \(\mu\) on \(E\), such that, in the class of all compact continua connecting \(a\) and \(b\) and not intersecting \(E\), the minimum of the weighted capacity in the external field \(\psi = -V^\mu\) is not attained.
logarithmic potential on the complex plane
scientific article; zbMATH DE number 6527089
scientific article