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

In a convex domain $\text{O}\subset {\mathbb{R}}^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$withadiscontinuouspinningterm$a$among$H^1(\O,\C)$-mapssubjecttoaDirichletboundarycondition$g\in H^{1/2}(\p\O,\S^1)$\mathrm{.}Thepinningterm$a:\R^3\to\R^*_+$takesaconstantvalue$b\in(0,1)$in$\o$,aninnerstrictlyconvexsubdomainof$\O$,and1outside$\o$. We prove energy estimates with various error terms depending on assumptions on $\text{O},\text{o}$ 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 $\pm 1$), 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_{\sigma(i)} where $\sigma$ is a minimal connection (computed w.r.t. a metric $d_{a^2}$) of the singularities of $g$ and $p_1,\mathrm{.}\mathrm{.}\mathrm{.}{p}_k$ are the positive (resp. $n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k$ the negative) singularities.