The Douglas problem for parametric double integrals
type: Item
instance of: scholarly article
title: The Douglas problem for parametric double integrals
zbMATH Open document ID: 1066.49012
author: Heiko von der Mosel
author: Matthias Kurzke
published in: Manuscripta Mathematica
publication date: 2003-03-27T00:00:00Z
review text: The main result of this paper is the existence of multiply connected, conformally parametrized minimizers of genus zero for a parametric semi-elliptic double integral spanned by a given system of Jordan curves. Consider parametric functionals \({\mathcal F}\) of the form \({\mathcal F}_ B(X):=\int_ B F(X, X_ u\wedge X_ v) dudv\), where \(F \in C^ 0({\mathbb R}^ n\times{\mathbb R}^ N)\), \(n\geq 2\), \(N=n(n-1)/2\), \(B\) is a domain in \({\mathbb R}^ 2\), and \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\). It is assumed that \(F\) satisfies the homogeneity condition and \(F\) is positive definite (there exist positive constants \(m_ 1\), \(m_ 2\) such that \(m_ 1| z| \leq F(x, z)\leq m_ 2| z| \) holds for all \((x, z)\in {\mathbb R}^ n\times {\mathbb R}^ N\)). Let \(\Gamma:=\langle\Gamma^ 1, \Gamma^ 2, ..., \Gamma^ k\rangle\) be a system of finitely many disjoint rectifiable closed Jordan curves in \({\mathbb R}^ n\). It is said that \(\Gamma\) satisfies the Douglas condition for \({\mathcal F}\) if and only if \(d(\Gamma)<d^{\dagger}(\Gamma)\) holds. Here, \(d(\Gamma)\) is the infimum of \({\mathcal F}_ B(X)\) among all discs \(B\) with \(k-1\) discs removed and all \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\) which map \(\partial B\) onto \(\Gamma\) weakly monotonically, and \(d^{\dagger}(\Gamma)\) is the infimum of \({\mathcal F}_ B(X)\) among all disconnected compact planer regions \(B\) with \(k\) boundary components and all \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\) which map \(\partial B\) onto \(\Gamma\) weakly monotonically. Under the assumption that \(F\) is semi-elliptic and \(\Gamma\) satisfies a chord-arc condition and \(F\) satisfies the Douglas condition for \({\mathcal F}\), it is proved that there exists a pair \((B, X)\) which attains \(d(\Gamma)\), and \(X\) is conformally parametrized. The main method of the proof is the Courant's classical approach for minimal…
reviewed by: Miyuki Koiso
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1889223
zbMATH Keywords: parametric integral
zbMATH Keywords: Douglas problem
zbMATH Keywords: isoperimetric inequality
MaRDI profile type: Publication
full work available at URL: s00229-002-0325-5
OpenAlex ID: W2011676051
DOI: 10.1007/S00229-002-0325-5
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description: scientific article; zbMATH DE number 1889223