On the Ginzburg-Landau energy with a weight vanishing at the boundary
The paper studies the behavior as \(\varepsilon \to 0\) of the minimizers of the Ginzburg--Landau energy with weight \[ E^w_\varepsilon (u)= {1\over 2} \int_G |\nabla u|^2+ {1\over {4\varepsilon^2}} \int_G (1-|u|^2)^2w \] in the class \(H^1_g(G)=\{u\in H^1(G;{\mathbb R}^2): u=g\) on \(\partial G\},\) where \(G\subset {\mathbb R}^2\) is a smooth and bounded domain and \(g:\partial G\to S^1\) is a smooth function with topological degree \(d>0.\) The weight \(w\) is of the type \[ w(z)=|z-z_0|^p+f(z), \] with \(f\in C^1(\overline{G}),\) \(f(z)=o(|z-z_0|^p),\) \(|\nabla f(z)|=o(|z-z_0|^{p-1})\) in a neighborhood of \(z_0\in \partial G\) and \(p>1.\)
scientific article; zbMATH DE number 1556811