Conformal welding for domains close to a disk
type: Item
instance of: scholarly article
title: Conformal welding for domains close to a disk
zbMATH Open document ID: 1276.30016
published in: Analysis and Mathematical Physics
publication date: 2012-12-04T00:00:00Z
review text: This interesting paper deals with conformal welding of starlike Jordan domains. Let \(D\) be a Jordan domain containing the origin and let \(f:{\mathbb D}\to D\) and \(F:{\mathbb D^*}\to\overline{\mathbb C}\setminus\overline D\) be the conformal mappings of the unit disk~\({\mathbb D}\) onto~\(D\) and of the exterior \({\mathbb D^*}:=\overline{\mathbb C}\setminus\overline{\mathbb D}\) of the unit disk onto the exterior of~\(D\), respectively, subject to the normalization \(f(0)=0\), \(f'(0)>0\), \(F(\infty)=\infty\), \(F'(\infty)>0\). Being extended by continuity to the unit circle these maps define a homeomorphism \(\gamma:\partial{\mathbb D}\to \partial{\mathbb D};\, e^{is}\mapsto e^{i\sigma}\) by the equation \(F(\gamma(e^{is}))=f(e^{is})\) for all \(s\in[0,2\pi]\). The homeomorphism \(\gamma\) is said to be the conformal welding associated to the Jordan domain~\(D\). The author studies conformal welding associated to starlike domains close to the unit disk. Let \(D\) be given in polar coordinate by \[ D:=\big\{\rho e^{i\psi}:\,\psi\in[0,2\pi],\,0\leq \rho\leq 1-\delta(\psi)\big\}, \] where \(\delta:{\mathbb R}\to(-\infty,1)\) is a twice differentiable \(2\pi\)-periodic function. Using a variational formula due to \textit{G. V. Siryk} [Usp. Mat. Nauk 11, No. 5(71), 57--60 (1956; Zbl 0074.06102)] and its analogue for the ``exterior'' map~\(F\) (Theorem~2), the author establishes the following asymptotic relation (Proposition 1) for the conformal welding \({e^{i\sigma}=\gamma(e^{is})}\), \(0\leq s\leq 2\pi\), associated to~\(D\): \[ s+h(s)=\sigma-h(\sigma)+ O(\epsilon^2)\text{ for all \(s\in[0,2\pi]\)} \] \[ \text{as }\epsilon:=\sup\{|\delta^{(k)}(\psi)|:\psi\in[0,2\pi],\,k=0,1,2\}\to+0, \] where \[ h(x):=\int_0^{2\pi}(\delta(\psi)-\delta(x))\cot\frac{\psi-x}{2}\,d\psi,\quad 0\leq x\leq2\pi. \] Similar formulas are obtained for domains \(D(\varphi,\alpha)\) bounded by \(l_1:=\big\{e^{i\theta}:\theta\in[\varphi,2\pi-\varphi]\big\}\) and the circular arc \(l_2\subset…
reviewed by: Pavel Gumenyuk
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 6112464
zbMATH Keywords: conformal welding
zbMATH Keywords: Loewner equation
zbMATH Keywords: parametric representation
zbMATH Keywords: conformal map
zbMATH Keywords: univalent function
zbMATH Keywords: variational formula
zbMATH Keywords: Löwner-Kufarev equation
author: Dmitri V. Prokhorov
MaRDI profile type: Publication
full work available at URL: s13324-011-0007-0
OpenAlex ID: W2035949633
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DOI: 10.1007/S13324-011-0007-0
description: scientific article; zbMATH DE number 6112464