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

Abstract

In a convex domain $Ø\subset {\mathbb{R}}^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_Ø|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(Ø,\C)$-maps subject to a Dirichlet boundary condition $g\in H^{1/2}(\pØ,§^1)$. The pinning term $a:{\mathbb{R}}^3\to {\mathbb{R}}_{+}^{\ast }$ takes a constant value $b\in (0,1)$ in $ø$, an inner strictly convex subdomain of $Ø$, and 1 outside $ø$. We prove energy estimates with various error terms depending on assumptions on $Ø,ø$ 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 $\pm 1$), vorticity defects are geodesics (computed w.r.t. a geodesic metric $d_{a^2}$ depending only on $a$) joining two paired singularities of $g$ $p_i & n_{σ(i)}$ where $\sigma$ 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. $n_1,...,n_k$ the negative) singularities.