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On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions. (English) Zbl 0809.35031
Summary: Minimizers \(u_ \varepsilon\) of the Ginzburg-Landau energy \(E_ \varepsilon\) defined by \[ E_ \varepsilon(u)= {\textstyle {1\over 2}} \int_ \Omega \Bigl\{ |\nabla u|^ 2+ {\textstyle {1\over {2\varepsilon^ 2}}} (1- | u|^ 2)^ 2 \Bigr\}dx \] on an arbitrary domain \(\Omega\subset \subset\mathbb{R}^ 2\) with smooth and boundary data \(g: \partial\Omega\to S^ 1\) as \(\varepsilon \to 0\) are shown to subconverge weakly in \(H^{1,p}\) for \(p<2\) and locally in \(H^{1,2}\) away from finitely many points \(x_ 1,\dots, x_ J\) to a smooth harmonic map \(u:\Omega \setminus\{ x_ 1,\dots, x_ J\}\to S^ 1\). The proof is based on simple comparison arguments. The result simplifies and extends previous work of Bethuel-Brezis-HĂ©lein for the same problem on a star-shaped domain.

35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations