On the symmetry of minimizing harmonic maps in \(N\) dimensions
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 \(n=2\), \(k=0\).
scientific article; zbMATH DE number 459226