Multiple surfaces of non-constant mean curvature
type: Item
instance of: scholarly article
title: Multiple surfaces of non-constant mean curvature
zbMATH Open document ID: 0814.53008
DOI: 10.1007/BF02571958
author: Norbert Jakobowsky
published in: Mathematische Zeitschrift
publication date: 1995-06-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174704
review text: Given some closed rectifiable Jordan curve in \(\mathbb{R}^ 3\) we prove existence of at least two solutions to the Plateau problem for surfaces of prescribed mean curvature parametrized over the unit disk provided the prescribed mean curvature function is sufficiently near (in \(L^ \infty\) sense) a suitable non-vanishing constant: Applying minimax methods in \(W^{1,2}\), we derive a mountain pass lemma for smooth closed Jordan curves and regular (non-constant) mean curvature functions. Proving a Palais Smale condition, we establish the existence of two distinct solutions in the smooth case. The full result is obtained by approximation. As an essential tool in both steps we have a precompactness result in \(W^{1,2}_{\text{loc}} \overline {(B_ 1(0)} \backslash \{q_ 1, \dots, q_ N\})\) for Palais Smale sequences.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 35J50
zbMATH DE Number: 737177
zbMATH Keywords: Plateau problem
zbMATH Keywords: prescribed mean curvature
zbMATH Keywords: mountain pass lemma
zbMATH Keywords: Palais Smale condition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 737177