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

Mickaël dos Santos (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

In a convex domain $\O\subset\R^3$, we consider the minimization of a $3D$-Ginzburg-Landau type energy $E_\v(u)=\frac{1}{2}\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ 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.

Domaines: Equations aux dérivées partielles [math.AP]

Mickaël dos Santos. Study of a $3D$ Ginzburg-Landau functional with a discontinuous pinning term. Nonlinear Analysis: Theory, Methods and Applications, 2012, 75 (17), pp.6275-6296. ⟨10.1016/j.na.2012.07.004⟩. ⟨hal-00578445v3⟩