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Radial symmetry of minimizers to the weighted Dirichlet energy

Published online by Cambridge University Press:  20 February 2020

Aleksis Koski
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland ([email protected])
Jani Onninen
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY13244, USA ([email protected])

Abstract

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

Type
Research Article
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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