Study of a 3D-Ginzburg-Landau functional with a discontinuous pinning term

In a convex domain Ω⊂R3, we consider the minimization of a 3D-Ginzburg-Landau type energy Eε(u)=12∫Ω|∇u|2+12ε2(a2−|u|2)2 with a discontinuous pinning term a among H1(Ω,C)-maps subject to a Dirichlet boundary condition g∈H1/2(∂Ω,S1). The pinning term a:R3→R+∗ takes a constant value b∈(0,1) in ω, an inner strictly convex subdomain of Ω, and 1 outside ω. We prove energy estimates with various error terms depending on assumptions on Ω,ω 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 da2 depending only on a) joining two paired singularities of gpi&nσ(i) where σ is a minimal connection (computed w.r.t. a metric da2) of the singularities of g and p1,…,pk are the positive (resp. n1,…,nk are the negative) singularities.