Microscopic renormalized energy for a pinned Ginzburg-Landau functional. The circular and degree one case

Mickaël dos Santos (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We study the vortex location for minimizers of a Ginzburg-Landau energy with a discontinuous constraint. The discontinuous constraint appears in the potential (a2 − |u|2)2. The function a is piecewise constant: it takes the value 0 < b < 1 in small disjoint domains (called inclusions) and 1 otherwise. It is proved, under some assumptions on the smallness of the inclusions and on their interdistances, that the vortices of minimizers are trapped inside the inclusions. Moreover the asymptotic location of the vortices inside an inclusion depends only on three parameters: the value b, the geometry of the inclusion and the number of vortices inside the inclusion. It is expected that, if an inclusion containing a unique vortex is a disk, then the asymptotic location of the vortex is the center of the inclusion. This article is dedicated to the proof of this expectation.

Mots clés: Ginzburg-Landau type energy; pinning; renormalized energy

Domaines: Equations aux dérivées partielles [math.AP]

Mickaël dos Santos. Microscopic renormalized energy for a pinned Ginzburg-Landau functional. The circular and degree one case. Calculus of Variations and Partial Differential Equations, 2015, 53 (1-2), pp.65-89. ⟨hal-00795405⟩