Topological sectors for Ginzburg-Landau energies
type: Item
instance of: scholarly article
title: Topological sectors for Ginzburg-Landau energies
zbMATH Open document ID: 0952.35049
DOI: 10.4171/RMI/264
author: Yanyan Li
published in: Revista Matemática Iberoamericana
publication date: 2001-01-04T00:00:00Z
full work available at URL: https://eudml.org/doc/39577
review text: 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 \(\Omega.\)
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35A17
zbMATH DE Number: 1419787
zbMATH Keywords: energy functional
zbMATH Keywords: topological degree
zbMATH Keywords: mountain-pass solution
zbMATH Keywords: critical point of functional
reviewed by: Igor V. Skrypnik
MaRDI profile type: Publication
OpenAlex ID: W1994692436
description: scientific article; zbMATH DE number 1419787