Continuity of the capacity in the class of capacitors with uniformly perfect plates
type: Item
instance of: scholarly article
title: Continuity of the capacity in the class of capacitors with uniformly perfect plates
zbMATH Open document ID: 0979.31005
published in: Doklady Mathematics
publication date: 2001-01-28T00:00:00Z
review text: Let \(\overline\mathbb{R}^n=\mathbb{R}^n\cup\{\infty\}\), and let comp \(\overline \mathbb{R}^n\) denote the space of all non-empty compact subsets of \(\overline \mathbb{R}^n\) equipped with the Hausdorff distance induced by the spherical distance in \(\overline \mathbb{R}^n\) [see \textit{K. Kuratowski}, `Topology, Volume 1', New York (1966; Zbl 0158.40802), p. 223]. A sequence of capacitors \(\{(E_k,F_k)\}\) is said to converge to the capacitor \((E,F)\) if \(E_k\to E\) and \(F_k\to F\) in comp \(\overline\mathbb{R}^n\) as \(k\to\infty\). The author proves that \[ \text{cap} (E_k,F_k)\to \text{cap}(E,F) \text{ as }k\to \infty, \] for capacitors whose plates belong to the class of uniformly perfect sets defined by \textit{C. Pommerenke} [Arch. Math. 32, 192-199 (1979; Zbl 0393.30005] in the plane.
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 1557252
zbMATH Keywords: capacity
zbMATH Keywords: uniformly perfect sets
author: V. V. Aseev
reviewed by: David A. Brannan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1557252