Elliptic capacity and its distortion under conformal mapping
type: Item
instance of: scholarly article
title: Elliptic capacity and its distortion under conformal mapping
zbMATH Open document ID: 1026.30023
DOI: 10.1007/BF02893086
author: Peter Duren
author: Reiner Kühnau
published in: Journal d'Analyse Mathématique
publication date: 2003-10-27T00:00:00Z
review text: For two points \(a,b\) in the extended complex plane \(\widehat{\mathbb C}\) let \[ [a,b]=\frac{\text{abs}{a-b}}{\text{abs}{1+\overline{a}b}}; \text{ (if } b=\infty, [a,\infty]=\text{abs}{a}^{-1}). \] This is the pseudoelliptic distance of \(a,b\); it does not satisfy the triangle inequality. If \(a\in \widehat{\mathbb C}\) and \(E\subset \widehat{\mathbb C}\), we define the antipodal point \(a^*\) and the antipodal set \(E^*\) by \(a^*=-1/\overline{a}\) and \(E^*=\{a^*:a\in E\}\). \textit{R. Kühnau} [ J. Reine Angew. Math. 234, 216-220 (1969; Zbl 0185.32802)], following ideas of M.Tsuji, developed a theory of elliptic transfinite diameter and elliptic capacity. In the paper under review the authors continue this development. They give an extremal length description of the elliptic capacity \(d_e(E)\) of \(E\subset \widehat{\mathbb C}\): \(d_e(E)=\exp (-\pi\lambda(C,C^*))\), where \(\lambda(C,C^*)\) is the extremal distance between the outer boundaries \(C, C^*\) of \(E,E^*\). They also develop a theory of elliptic Robin capacity in analogy with the theories of Robin and hyperbolic Robin capacities (see e.g. the articles by \textit{P. Duren} [Proceedings of the 3rd CMFT conference on computational methods and function theory 1997, Nicosia, Cyprus, October 13-17, 1997. Singapore: World Scientific. Ser. Approx. Decompos. 11, 177-190 (1999; Zbl 0939.30016)] and by \textit{P. Duren} and \textit{J. Pfaltzgraff} [J. Anal. Math. 78, 205-218 (1999; Zbl 0974.31001)]). Finally, they use a variational method to characterize the elliptic Robin capacity by an extremal property of conformal mappings.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 1997293
zbMATH Keywords: elliptic capacity
zbMATH Keywords: conformal mapping
zbMATH Keywords: elliptic Robin capacity
zbMATH Keywords: extremal distance
zbMATH Keywords: elliptic Neumann kernel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02893086
OpenAlex ID: W1976382072
description: scientific article; zbMATH DE number 1997293