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Radial symmetry of minimizers to the weighted Dirichlet energy

Part of: Elliptic equations and systems Geometric function theory

Published online by Cambridge University Press:  20 February 2020

Aleksis Koski  and
Jani Onninen
Aleksis Koski
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland (aleksis.t.koski@jyu.fi)
Jani Onninen
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY13244, USA (jkonnine@syr.edu)
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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.

Keywords

Variational integralsHarmonic mappingsEnergy-minimal deformations

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 169 - 186
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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