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 .T he 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.