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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-Hélein for the same problem on a star-shaped domain.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B25 Singular perturbations in context of PDEs 35J20 Variational methods for second-order elliptic equations
##### Keywords:
Ginzburg-Landau energy