Capacity of a compact set in a logarithmic potential field
type: Item
instance of: scholarly article
title: Capacity of a compact set in a logarithmic potential field
zbMATH Open document ID: 1335.31001
DOI: 10.1134/S0081543815060218
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2016-01-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 6527089
zbMATH Keywords: logarithmic potential on the complex plane
zbMATH Keywords: equilibrium problems
zbMATH Keywords: weighted capacity
author: Viktor I. Buslaev
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6527089