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# Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

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Volume 12, Issue 4

# Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions

Sascha Eichmann
• Corresponding author
• Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120,39016 Magdeburg, Germany
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Hans-Christoph Grunau
Published Online: 2017-08-12 | DOI: https://doi.org/10.1515/acv-2016-0038

## Abstract

In this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below $4\pi$. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x-axis. By working below the energy threshold of $4\pi$ and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

MSC 2010: 49Q10; 53C42; 34B30; 34C20; 35J62; 34L30

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## About the article

Received: 2016-08-25

Accepted: 2017-07-27

Published Online: 2017-08-12

Published in Print: 2019-10-01

Citation Information: Advances in Calculus of Variations, Volume 12, Issue 4, Pages 333–361, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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[2]
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