Renormalized Ginzburg-Landau energy and location of near boundary vortices
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1.
Department of Mathematics, Penn State University, University Park, State College, PA 16802
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Mathematical Division, B. I. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 61103
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3.
Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907
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Received:
01 April 2011
Revised:
01 December 2011
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Primary: 58E50, 35J20; Secondary: 49J10.
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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.
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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