Generalized condensers and the asymptotics of their capacities under degeneration of some plates
type: Item
instance of: scholarly article
title: Generalized condensers and the asymptotics of their capacities under degeneration of some plates
zbMATH Open document ID: 1241.31003
author: V. N. Dubinin
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-08-18T00:00:00Z
review text: Let us investigate a Jordan domain \(B\) in the complex plane \(\mathbb C\), and denote by \(\overline B\) (resp. \(\partial B\)) the closure of \(B\) (resp. topological boundary of \(B\)) in \(\overline{\mathbb C}\) (\(\mathbb C\cup\infty\), the Riemann sphere). Topology in \(\overline B\) is always understood to be that induced by \(\overline{\mathbb C}\) on this set. A \textit{condenser} \(C\) (\(\subset \overline B\)) is a triple \((B,\xi,\Delta)\), in which, \(E:=\{E_k\}_{k=1}^n\) is a collection of pairwise disjoint sets \(E_k\) (\(k=1,\dots,n\)), closed in \(\overline B\), \(\Delta=(\delta_k)_{k=1}^n\) is a set of \(n\) given real numbers. We denote it by \(C:= (B,\xi,\Delta)\) and call \(\xi\) the set of plates of the condenser \(C\). The \textit{capacity} of the condenser \(C\), denoted by \(\text{Cap}(C)\), is defined as the greatest lower bound (w.r.t. \(v\)) of the following Dirichlet integrals, \[ I (v; B) = \iint_B |\nabla v|^2 \,dx\,dy ,\quad v\in\mathcal F, \] where \(\mathcal F\) denotes the class of functions continuous in \(\overline B\), locally Lipschitz in \(B\) except of a finite number of points and furthermore constant and equal to \(\delta_k\) on \(E_k\) (\(k = 1,\dots, n\)). \(\mathcal F\) is called the class of \textit{admissible functions} in \(B\). In this paper the author continues his study (with others) on the asymptotic behaviour of the condenser's capacity in case some of its plates (including plates on \(\partial B\)) degenerate to a finite number of points. Under suitable conditions the author proves, grosso modo from Theorem 1 to Theorem 6 of Sect. 1, that \(\text{Cap}(\cdot)\) is indeed a conformal invariant Choquet's capacity. Sect. 2 is mainly reserved to the investigation of Robin's function and its analog with a pole at the boundary, and also to the notion of Robin's capacity. The asymptotic formula (formula 7) for \(\text{Cap}(\cdot)\) is given in the last Sect. 3 of the paper. The author notes that: 1) The formula 7 impli…
reviewed by: Xuan Loc Nguyen
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 5312791
zbMATH Keywords: coformal capacity
MaRDI profile type: Publication
full work available at URL: s10958-005-0319-4
OpenAlex ID: W2044245776
cites work: Invariant spaces and traces of holomorphic functions on the skeletons of classical domains
cites work: Confromal invariants and function-theoretic null-sets
cites work: Q3291667
cites work: Symmetrization in the geometric theory of functions of a complex variable
cites work: Reduced moduli of open sets in the theory of analytic functions
cites work: Q4700570
cites work: Q4396659
cites work: On the reflection principle and some elementary ratios of conformal radii
cites work: On the computation of modules of long quadrilaterals
cites work: Asymptotics of the module of a degenerating condenser and some of their applications
cites work: The reduced module of the complex sphere
DOI: 10.1007/S10958-005-0319-4
description: scientific article; zbMATH DE number 5312791