Large solutions for harmonic maps in two dimensions
type: Item
instance of: scholarly article
title: Large solutions for harmonic maps in two dimensions
zbMATH Open document ID: 0532.58006
DOI: 10.1007/BF01210846
author: Jean-Michel Coron
author: Haïm Brezis
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: The authors establish the existence of local (but not absolute) minimum of the energy functional \(E\) for the set \({\mathcal E}\) of maps \(u\in H^ 1(\Omega;\mathbb S^ 2)\) defined on the unit disk \(\Omega \subset {\mathbb R}^ 2\) into the Euclidean 2-sphere \(S^ 2\) (or a Riemannian surface homeomorphic to \(S^ 2)\) satisfying a nonconstant boundary condition \(u=\gamma\) on \(\partial \Omega\). (Nonexistence of nontrivial \(u\) for \(\gamma =\text{const}\) has been shown by \textit{L. Lemaire} [J. Differ. Geom. 13, 51--78 (1979; Zbl 0388.58003)]. The proof is based on considering the functional \(Q(u)=\frac{1}{4\pi}\int_{\Omega}u\cdot u_ x\wedge u_ y\) and showing that at least one of the two infima \(I_{\pm}=\text{Inf}\{E(u)\mid u\in {\mathcal E},\quad Q(u)-Q(\underline u)=\pm 1\}\) is achieved, where \(\underline u\in {\mathcal E}\) such that \(E(\underline u)=\text{Inf}\{E(u)\mid u\in {\mathcal E}\}.\) A detailed study of the case when \(\gamma\) is a small circle on \(S^ 2\) is also given.
reviewed by: Gábor Tóth
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3844673
zbMATH Keywords: energy functional
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Riemannian surface
zbMATH Keywords: nonconstant boundary condition
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01210846
OpenAlex ID: W2010532103
description: scientific article; zbMATH DE number 3844673