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

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.