The Plateau problem in unbounded domains
type: Item
instance of: scholarly article
title: The Plateau problem in unbounded domains
zbMATH Open document ID: 0534.49027
author: Elisabetta Barozzi
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1983-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: This paper is concerned with a problem of isoperimetric type: minimize \[ {\mathcal F}(E)=\int_{\Omega}| D\phi_ E| +\int | \phi_ E- \phi_{\Gamma}| dH^{n-1} \] in the class \({\mathcal E}={\mathcal E}_ v=\{E\subset \Omega:H^ n(E)=V\},\) where \(\Omega \subset {\mathbb{R}}^ n\) is open, \(\partial \Omega\) is Lipschitzian, \(\Gamma\subset \partial \Omega\), and V is a fixed real number with \(0<V<H^ n(\Omega)\). The difficulty lies in the fact that \(\Omega\) is not assumed bounded nor is it assumed that \(H^ n(\Omega)<\infty\). Two additional hypotheses are introduced: (i) \(\Gamma\) is a bounded set, (ii) There exists a sphere \(B\subset \Omega\) with \(H^ n(B)=V\). The hypothesis (ii) allows the construction of comparison sets which give useful estimates because of the isoperimetric property of spheres. The author is then able to construct a bounded minimizing sequence and prove the existence of a solution to the problem.
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 3848111
zbMATH Keywords: isoperimetric property
zbMATH Keywords: sets of finite perimeter
zbMATH Keywords: minimizing sequence
reviewed by: Harold Parks
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3848111