Study of a -Ginzburg–Landau functional with a discontinuous pinning term

• Published in: Nonlinear analysis Vol. 75; no. 17; pp. 6275 - 6296
• Main Author: Dos Santos, Mickaël
• Format: Journal Article
• Language: English
• Published: 01.11.2012
• Subjects: Computation; Defects; Dirichlet problem; Optimization; Pinning; Singularities; Three dimensional; Vorticity
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2012.07.004

In a convex domain O ? R 3 , we consider the minimization of a 3 D -Ginzburg-Landau type energy E [e] (u) = 1 2 ? O | ? u | 2 + 1 2 [e] 2 (a 2 - | u | 2) 2 with a discontinuous pinning term a among H 1 (O , C) -maps subject to a Dirichlet boundary condition g ? H 1 / 2 (? O , S 1) . The pinning term a : R 3 ? R + * takes a constant value b ? (0 , 1) in ? , an inner strictly convex subdomain of O , and 1 outside ? . We prove energy estimates with various error terms depending on assumptions on O , ? and g . In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of g (the singularities are polarized and quantified by their degrees which are +/- 1), vorticity defects are geodesics (computed w.r.t. a geodesic metric d a 2 depending only on a) joining two paired singularities of g p i & n s (i) where s is a minimal connection (computed w.r.t. a metric d a 2) of the singularities of g and p 1 , ... , p k are the positive (resp. n 1 , ... , n k are the negative) singularities.