Topological sectors for Ginzburg-Landau energies
The energy level set \[ E^{\Lambda}_\varepsilon=\Biggl\{u\in H^1(\Omega,\mathbb{R}^2): E_\varepsilon(u)\equiv\frac 12\int_\Omega |\nabla u|^2 dx+\frac{1}{4\varepsilon^2}\int_\Omega (1-|u|^2)^2 dx < \Lambda\Biggr\} \] for the Ginzburg-Landau functional \(E_\varepsilon(u)\) corresponding to the annulus \(\Omega=\{x\in \mathbb{R}^2:\frac 14<|x|<1\}\) is studied. It is proved that for sufficiently small \( \varepsilon \) the degree map \(\chi:E^\Lambda_\varepsilon\to \mathbb{Z}\) is constant inside each connected component of \(E^\Lambda_\varepsilon.\) Under some condition on the topological sectors \(\text{top}_n(E^\Lambda_\varepsilon)=\{u\in E^\Lambda_\varepsilon:\chi(u)=n\}\) the existence of mountain-pass type critical points of \(E_\varepsilon\) is established. Analogous results are proved for the gauge-covariant Ginzburg-Landau model and for more general domains \(\Omega.\)
scientific article; zbMATH DE number 1419787