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

Article type: Research Article

Authors: Misiats, Oleksandr

Affiliations: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. E-mail: omisiats@purdue.edu

Abstract: 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.

Keywords: Ginzburg–Landau minimizers, semi-stiff boundary conditions, approximate bulk degree

DOI: 10.3233/ASY-141227

Journal: Asymptotic Analysis, vol. 89, no. 1-2, pp. 37-61, 2014