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Advances in Calculus of Variations

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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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/ Hans-Christoph Grunau
Published Online: 2017-08-12 | DOI: https://doi.org/10.1515/acv-2016-0038


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 4π. 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 4π 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.

Keywords: Dirichlet boundary conditions; Willmore surfaces of revolution; elastic curves; projectability

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, DOI: https://doi.org/10.1515/acv-2016-0038.

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