Local minimizers for the Ginzburg-Landau energy
The authors consider the Ginzburg-Landau system: \[ -\Delta u= {1\over\varepsilon^2} (1-|u|^2)u\quad\text{in }\Omega,\quad u=g\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^2\), \(u:\overline\Omega\to \mathbb{R}^2\), \(g:\partial\Omega\to S^1\) is smooth and \(\varepsilon> 0\) is a parameter. Associated to problem (1) is the energy functional \[ E_\varepsilon(u)= {1\over 2} \int_\Omega|\nabla u|^2+{1\over 4\varepsilon^2} \int_\Omega(1-|u|^2)^2,\tag{2} \] critical points of which in \(H_g'= \{u\in H^1(\Omega,\mathbb{R}^2):u|_{\partial \Omega}= g\}\) correspond precisely to solutions of (1). In the case \(\Omega\) starshaped, for any sequence \(\{\varepsilon_n\}\) approaching zero and any global minimizer \(u_n\) of (2) (for \(\varepsilon= \varepsilon_n\)) there exists a subsequence which has a limit \[ u^*(x)= {(x- a_1)\over|x-a_1|}\cdots {(x- a_d)\over|x-a_d)} e^{i\psi(x)},\tag{3} \] where \(\psi\) is a real-valued function and satisfies \(\Delta\psi= 0\) in \(\Omega\), \(u^*= g\) on \(\partial\Omega\). Here \(a_1,\dots, a_d\in\Omega\) and \(d= \deg(g,\partial\Omega)\) is assumed to be nonnegative. The convergence is smooth in compact subsets of \(\overline\Omega\setminus\{a_1\cdots c_d\}\) and the singularities \(a_1,\dots, a_d\) correspond to global minimizers of the renormalized energy functionals, \(W:\Omega^d\to \mathbb{R}\), defined as \[ W(a_1,\dots, a_d)= -\pi \sum_{i\neq j}\log|a_i- a_j|+{1\over 2} \int_{\partial\Omega} \Phi(g\times g_\tau)- \pi\sum^d_{i= 1} R(a_i).\tag{4} \] The main result is the following: Let \(\Lambda\) be an open subset of \(\Omega^d\) such that \(\overline\Lambda\subset \Omega^d\) and \[ \inf_\Lambda W<\inf_{\partial\Lambda} W, \] where \(W\) is given by (4). Then any sequence \(\varepsilon_n\to 0\) possesses a subsequence also denoted by \(\varepsilon_n\) such that there is a family of solutions \(u_{\varepsilon_n}\) to (1) and a point \((a_1,\dots, a_d)\in\Lambda\) with \[ W(a_1,\dots, a_d)= \inf_\Lambda W \] such that \(u_{\varepsilon_n}\to u^*\) uniformly in the \(H^1\) sense on compact subsets of \(\overline\Omega\setminus\{a_1,\dots, a_d\}\), and \(u_{\varepsilon_n}\) is a local minimizer of \(E_{\varepsilon_n}\).
scientific article; zbMATH DE number 1374307