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energy-minimal deformations

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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1 results

Radial symmetry of minimizers to the weighted Dirichlet energy
Part of
- Elliptic equations and systems
- Geometric function theory
Aleksis Koski, Jani Onninen
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 1 / February 2021

Published online by Cambridge University Press:

20 February 2020, pp. 169-186

Print publication:

February 2021
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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.