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Advances in Calculus of Variations

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Volume 10, Issue 4 (Oct 2017)

Energy and area minimizers in metric spaces

Alexander Lytchak / Stefan Wenger
  • Corresponding author
  • Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland
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Published Online: 2016-09-22 | DOI: https://doi.org/10.1515/acv-2015-0027


We show that in the setting of proper metric spaces one obtains a solution of the classical 2-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hölder exponents of some area-minimizing discs.

Keywords: Plateau’s problem; energy minimizers; Sobolev maps; quasi-convex areas; Hölder regularity; metric spaces

MSC 2010: 49Q05; 52A38


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About the article

Received: 2015-07-17

Revised: 2016-08-24

Accepted: 2016-09-03

Published Online: 2016-09-22

Published in Print: 2017-10-01

Funding Source: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award identifier / Grant number: 153599

The second author was partially supported by Swiss National Science Foundation Grant 153599.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0027.

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