Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values
type: Item
instance of: scholarly article
title: Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values
zbMATH Open document ID: 0808.58017
DOI: 10.1007/BF02567598
author: Ernst Kuwert
published in: Manuscripta Mathematica
publication date: 1995-03-16T00:00:00Z
full work available at URL: https://eudml.org/doc/155945
review text: Let \(S \subset \mathbb{R}^ n\) be an embedded smooth 2-sphere, and let \(\Sigma\) be a compact Riemann surface with nonempty boundary. For a given boundary datum \(\nu \in \text{Lip} (\partial \Sigma, S)\) consider the class \(E = \{u \in W^{1,2} (\Sigma, \mathbb{R}^ n) : u|_{\partial \Sigma} = \nu\), \(u(z) \in S\) for a.e. \(z \in \Sigma\}\). Let \(E_ k = \{u \in E: \text{deg}(u,u_ 0) = k\}\), where \(u_ 0 \in E\) is a fixed map, \(\text{deg}(u,u_ 0)\) is the degree of \(u\) relative \(u_ 0\). There is considered the question: \[ \text{Is } \lambda_ k = \inf \{E(u) = 2^{- 1} \cdot \int_ \Sigma (| u_ x|^ 2 + | u_ y|^ 2 dx dy : u \in E_ k\} \text{ attained?}\tag{1} \] If \(E\) contains a holomorphic (antiholomorphic) map \(u^ +\) (\(u^ -\) respectively), then let \(\text{deg}(u^ +, u_ 0) = k^ +\) \((\text{deg} (u^ -, u_ 0) = k^ -\)). Otherwise set \(k^ + = + \infty\), \(k^ - = - \infty\). The following result is proved. Theorem. \(\lambda_ k\) is attained if and only if \(k^ - \leq k \leq k^ +\). The theorem reduces (1) to a problem in complex function theory.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 599662
zbMATH Keywords: minimizer of the energy
zbMATH Keywords: relative degree
zbMATH Keywords: holomorphic map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 599662