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Advances in Calculus of Variations

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Harmonic maps between two concentric annuli in 𝐑3

David Kalaj
  • Corresponding author
  • Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro
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Published Online: 2019-07-12 | DOI: https://doi.org/10.1515/acv-2018-0074

Abstract

Given two annuli 𝔸(r,R) and 𝔸(r,R), in 𝐑3 equipped with the Euclidean metric and the weighted metric |y|-2, respectively, we minimize the Dirichlet integral, i.e., the functional

[f]=𝔸(r,R)Df2|f|2,

where f is a homeomorphism between 𝔸(r,R) and 𝔸(r,R), which belongs to the Sobolev class 𝒲1,2. The minimizer is a certain generalized radial mapping, i.e., a mapping of the form f(|x|η)=ρ(|x|)T(η), where T is a conformal mapping of the unit sphere onto itself and ρ(t)=R(rR)R(r-t)(R-r)t. It should be noticed that, in this case, no Nitsche phenomenon occurs.

Keywords: Minimizers; Nitsche phenomenon; annuli

MSC 2010: 31A05; 42B30

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About the article


Received: 2018-11-08

Revised: 2019-05-05

Accepted: 2019-06-06

Published Online: 2019-07-12


Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2018-0074.

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