The necessary conditions for the existence of local Ginzburg–Landau minimizers with prescribed degrees on the boundary

• Published in: Asymptotic analysis Vol. 89; no. 1-2; pp. 37 - 61
• Main Author: Misiats, Oleksandr
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.09.2014
• Subjects: Ginzburg–Landau minimizers; semi-stiff boundary conditions; approximate bulk degree
• ISSN: 0921-7134; 1875-8576
• DOI: 10.3233/ASY-141227

We study the minimization problem for simplified Ginzburg–Landau functional in doubly connected domain. This minimization problem is a subject to “semi-stiff” boundary conditions: |u|=1 and prescribed degrees p and q on the outer and inner boundaries respectively. Following the work of Berlyand and Rybalko [J. Eur. Math. Soc. 12 (2010), 1497–1531], we additionally prescribe the degree in the bulk (approximate bulk degree) to be d. The work [J. Eur. Math. Soc. 12 (2010), 1497–1531] established the sufficient conditions on the existence of Ginzburg–Landau minimizers, given in terms of p, q and d. The present work complements the result of [J. Eur. Math. Soc. 12 (2010), 1497–1531] by providing the necessary conditions for the existence of nontrivial (nonconstant) minimizers.