Local solutions to a free boundary problem for the Willmore functional

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• 作者：Roberta Alessandroni ; Ernst Kuwert
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：55
• 期：2
• 全文大小：590 KB
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• 作者单位：Roberta Alessandroni (1)
  Ernst Kuwert (1)
  
  1. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104, Freiburg, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We consider a free boundary problem for the Willmore functional \(\mathcal{W}(f) = \frac{1}{4} \int _\Sigma H^2\,d\mu _f\). Given a smooth bounded domain \(\Omega \subset {\mathbb R}^3\), we construct Willmore disks which are critical in the class of surfaces meeting \(\partial \Omega \) at a right angle along their boundary and having small prescribed area. Using rescaling and the implicit function theorem, we first obtain constrained solutions with prescribed barycenter on \(\partial \Omega \). We then study the variation of that barycenter.