Remarks on approximate harmonic maps
type: Item
instance of: scholarly article
title: Remarks on approximate harmonic maps
zbMATH Open document ID: 0831.58017
DOI: 10.1007/BF02566002
author: Yumnei Chen
author: Fanghua Lin
published in: Commentarii Mathematici Helvetici
publication date: 1996-02-22T00:00:00Z
full work available at URL: https://eudml.org/doc/140364
review text: Given Riemannian manifolds \(M\) and \(N\) with \(N\) embedded in some space \(\mathbb{R}^k\), the Dirichlet energy for mappings \(M\to N\) is approximated by the functional \(\int_M |\nabla u|^2+ {1\over \varepsilon^2} d^2 (u, N) dx\) \((0< \varepsilon< 1)\) defined on spaces of mappings \(u: M\to \mathbb{R}^k\) where \(d( \cdot, N)\) measures the distance to the manifold \(N\). For the special choice \(N= S^1\) and for two-dimensional domains \(M\) the authors prove uniform estimates for minimizers of the above energy in the class \(H^1 (M, \mathbb{C})\) with prescribed boundary data \(g\) such that \(|g(x) |=1\) and \(\deg (g, \partial M) =0\). They apply these results to give a short proof of some theorems due to \textit{F. Bethuel}, \textit{H. Brezis} and \textit{F. Helein} [Calc. Var. Partial Differ. Equ. 1, No. 2, 123-148 (1993)]\ concerning asymptotic limits for the Ginzburg-Landau model of scalar fields. Moreover, a new proof of the Schoen-Uhlenbeck regularity theorem [\textit{R. Schoen} and \textit{K. Uhlenbeck}, J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)]\ is given and it is shown how to apply this theorem to obtain uniqueness for the harmonic map heat flow.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 791088
reviewed by: M. Fuchs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 791088