Study of a 3D Ginzburg-Landau functional with a discontinuous pinning term - Santos, Mickaël Dos (arXiv.org, 2012)
In a convex domain \(\O\subset\R^3\), we consider the minimization of a 3D-Ginzburg-Landau type energy \(E_\v(u)=1/2\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2\) with a discontinuous pinning term \(a\) among \(H^1(\O,\C)\)-maps subject to a Dirichlet boundary condition \(g\in H^{1/2}(\p\O,\S^1)\). The pinning term \(a:\R^3\to\R^*_+\) takes a constant value \(b\in(0,1)\) in \(\o\), an inner strictly convex subdomain of \(\O\), and 1 outside \(\o\). We prove energy estimates with various error terms depending on assumptions on \(\O,\o\) 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 \(p_1,...p_k\) are the positive (resp. \(n_1,...,n_k\) the negative) singularities.