Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality

We consider the Ginzburg–Landau energy \(E_{\varepsilon }\) $E_{\epsilon }$ for \(\mathbb {R}^M\) ${\mathbb{R}}^M$ -valued maps defined in a cylinder shape domain \(B^N\times (0,1)^n\) $B^N\times {(0,1)}^n$ satisfying a degree-one vortex boundary condition on \(\partial B^N\times (0,1)^n\) $\partial B^N\times {(0,1)}^n$ in dimensions \(M\ge N\ge 2\) $M\ge N\ge 2$ and \(n\ge 1\) $n\ge 1$ . The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if \(N\ge 7\) $N\ge 7$ , then for every \({\varepsilon }>0\) $\epsilon >0$ , there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution \(u_{\varepsilon }(x,z)=(f_{\varepsilon }(|x|) \frac{x}{|x|}, 0_{\mathbb {R}^{M-N}})\) $u_{\epsilon }(x,z)=(f_{\epsilon }(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N}})$ , \(\forall x\in B^N\) $\forall x\in B^N$ that is invariant in \(z\in (0,1)^n\) $z\in {(0,1)}^n$ . If \(2\le N \le 6\) $2\le N\le 6$ and \(M\ge N+1\) $M\ge N+1$ , then the following dichotomy occurs between escaping and non-escaping solutions: there exists \({\varepsilon }_N>0\) ${\epsilon }_N>0$ such that if \({\varepsilon }\in (0, {\varepsilon }_N)\) $\epsilon \in (0,{\epsilon }_N)$ , then every global minimizer is an escaping radially symmetric vortex sheet solution of the form \(R \tilde{u}_{\varepsilon }\) $R{\tilde{u}}_{\epsilon }$ where \(\tilde{u}_{\varepsilon }(x,z)=(\tilde{f}_{{\varepsilon }}(|x|) \frac{x}{|x|}, 0_{\mathbb {R}^{M-N-1}}, g_{{\varepsilon }}(|x|))\) ${\tilde{u}}_{\epsilon }(x,z)=({\tilde{f}}_{\epsilon }(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N-1}},g_{\epsilon }(|x|))$ is invariant in z-direction with \(g_{\varepsilon }>0\) $g_{\epsilon }>0$ in (0, 1) and \(R\in O(M)\) $R\in O(M)$ is an orthogonal transformation keeping invariant the space \(\mathbb {R}^N\times \{0_{\mathbb {R}^{M-N}}\}\) ${\mathbb{R}}^N\times \{0_{{\mathbb{R}}^{M-N}}\}$ ; if \({\varepsilon }\ge {\varepsilon }_N\) $\epsilon \ge {\epsilon }_N$ , then the non-escaping radially symmetric vortex sheet solution \(u_{\varepsilon }(x,z)=(f_{\varepsilon }(|x|) \frac{x}{|x|}, 0_{\mathbb {R}^{M-N}})\) $u_{\epsilon }(x,z)=(f_{\epsilon }(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N}})$ , \(\forall x\in B^N, z\in (0,1)^n\) $\forall x\in B^N,z\in {(0,1)}^n$ is the unique global minimizer; moreover, there are no bounded escaping solutions in this case. We also discuss the problem of vortex sheet \({\mathbb S}^{M-1}\) ${\mathbb{S}}^{M-1}$ -valued harmonic maps.