Renormalized Ginzburg-Landau energy and location of near boundary vortices

Leonid Berlyand; Volodymyr Rybalko; Nung Kwan Yip; Leonid Berlyand; Volodymyr Rybalko; Nung Kwan Yip

doi:10.3934/nhm.2012.7.179

Networks and Heterogeneous Media

2012, Volume 7, Issue 1: 179-196. doi: 10.3934/nhm.2012.7.179

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Renormalized Ginzburg-Landau energy and location of near boundary vortices

1.
Department of Mathematics, Penn State University, University Park, State College, PA 16802
2.
Mathematical Division, B. I. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 61103
3.
Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907

Received: 01 April 2011 Revised: 01 December 2011
Primary: 58E50, 35J20; Secondary: 49J10.

We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.
- Ginzburg-Landau energy,
- renormalized energy,
- near boundary vortices
Citation: Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices[J]. Networks and Heterogeneous Media, 2012, 7(1): 179-196. doi: 10.3934/nhm.2012.7.179

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Abstract

We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.

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