On the symmetry of minimizing harmonic maps in \(N\) dimensions
type: Item
instance of: scholarly article
title: On the symmetry of minimizing harmonic maps in \(N\) dimensions
zbMATH Open document ID: 0780.58012
author: Itai Shafrir
author: Étienne Sandier
published in: Differential and Integral Equations
publication date: 1994-02-02T00:00:00Z
review text: Let \(\Omega\) be a domain in \(\mathbb{R}^{k+n}\). We say that \(\Omega\) is \(S^{n-1}\)-symmetric (or symmetric) if it can be written, for some domain \(\omega\) in \(\mathbb{R}_ +^{k+1}\), as \[ \Omega=\bigl\{(x,ry):(x,r)\in\omega,\;y\in S^{n-1}\bigr\}, \] where \(\mathbb{R}_ +^{k+1}=\bigl\{(x,r):x\in\mathbb{R}^ k,\;r\in\mathbb{R}_ +\bigr\}\). A map \(u\) defined on such a domain, with values in \(S^ n\), is said to be symmetric if there exists a function \(\varphi:\omega\to\mathbb{R}\) such that \[ u(x,ry)=\bigl(y\cos\varphi(x,r),\sin\varphi(x,r)\bigr), \quad\forall(x,ry)\in\Omega. \] A minimizing harmonic map (or a minimizer) for boundary data \(\Psi\in H^{1/2}(\partial\Omega,S^ n)\) is a map \({u:\Omega\to S^ n}\) whose Dirichlet energy \(E(u)=\int_ \Omega|\nabla u|^ 2dV\) is minimal among the maps \(v\in H^ 1(\Omega,S^ n)\) satisfying \(v=\Psi\) on \(\partial\Omega\). The symmetry problem for minimizing harmonic maps can be stated as follows: is a minimizer for symmetric boundary data \(\Psi:\partial\Omega\to S^ n\) necessarily symmetric? Although the answer to this question is negative in general, it follows from results of W. Jäger and H. Kaul that for a symmetric \(\Psi:\partial\Omega\to S^ n\) with image lying in a compact subset of a hemisphere of \(S^ n\), there is a unique minimizer, which is in particular symmetric. Our main theorem states that this last result still holds, with an obvious modification of the uniqueness statement, when we only assume that \(\Psi(\partial\Omega)\) is contained in a closed hemisphere of \(S^ n\); i.e., in \(S^ n_ +=S^ n\cap\{x_{n+1}\geq 0\}\) or \(S^ n_ - =S^ n\cap\{x_{n+1}\leq 0\}\). We remark that this result was known for the case \(n=2\), \(k=0\).
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 459226
zbMATH Keywords: symmetry
zbMATH Keywords: minimizing harmonic maps
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 459226