Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Astala, Kari / Colding, Tobias / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Finster, Felix / Gianazza, Ugo / Gursky, Matthew / Hardt, Robert / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / McCann, Robert / Mingione, Giuseppe / Nystrom, Kaj / Pacard, Frank / Preiss, David / Riviére, Tristan / Schaetzle, Reiner / Silvestre, Luis

4 Issues per year

IMPACT FACTOR 2016: 1.182

CiteScore 2016: 0.78

SCImago Journal Rank (SJR) 2016: 1.277
Source Normalized Impact per Paper (SNIP) 2016: 0.881

Mathematical Citation Quotient (MCQ) 2016: 0.83

See all formats and pricing
More options …
Volume 4, Issue 1


Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

Anna Dall'Acqua / Steffen Fröhlich
  • Institut für Mathematik, Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany.
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Hans-Christoph Grunau / Friedhelm Schieweck
Published Online: 2010-08-23 | DOI: https://doi.org/10.1515/acv.2010.022


We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the “length” of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles.

These analytical investigations are complemented by presenting a numerical algorithm, based on C 1-elements, and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.

Keywords.: Dirichlet boundary conditions; Willmore surfaces of revolution

About the article

Received: 2009-02-17

Revised: 2010-06-07

Published Online: 2010-08-23

Published in Print: 2011-01-01

Citation Information: Advances in Calculus of Variations, Volume 4, Issue 1, Pages 1–81, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv.2010.022.

Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Sascha Eichmann and Hans-Christoph Grunau
Advances in Calculus of Variations, 2017, Volume 0, Number 0
Sascha Eichmann and Amos Koeller
The Journal of Geometric Analysis, 2017, Volume 27, Number 1, Page 618
Rainer Mandel
Calculus of Variations and Partial Differential Equations, 2015, Volume 54, Number 4, Page 3905
Sascha Eichmann
The Journal of Geometric Analysis, 2016, Volume 26, Number 4, Page 2563
James McCoy and Glen Wheeler
Mathematische Annalen, 2013, Volume 357, Number 4, Page 1485
Anna Dall’Acqua
Annals of Global Analysis and Geometry, 2012, Volume 42, Number 3, Page 411
Rustum Choksi and Marco Veneroni
Calculus of Variations and Partial Differential Equations, 2012
Anna Dall’Acqua, Klaus Deckelnick, and Glen Wheeler
Calculus of Variations and Partial Differential Equations, 2013, Volume 48, Number 3-4, Page 293
Matthias Bergner, Anna Dall’Acqua, and Steffen Fröhlich
Journal of Geometric Analysis, 2013, Volume 23, Number 1, Page 283
Matthias Bergner and Lars Schäfer
Journal of Geometry and Physics, 2011, Volume 61, Number 10, Page 1985

Comments (0)

Please log in or register to comment.
Log in