21734 articles – 15570 references  [version française]

hal-00578445, version 2

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

Mickaël Dos Santos (, ) 1

Nonlinear Analysis: Theory, Methods and Applications 75, 17 (2012) 6275-6296

Abstract: 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.

• 1:  Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
• Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout

• hal-00578445, version 2
• oai:hal.archives-ouvertes.fr:hal-00578445
• From:
• Submitted on: Monday, 5 March 2012 18:35:11
• Updated on: Tuesday, 28 August 2012 18:06:24