The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: the non-zero degree case
Mickael Dos santos
49K2035J6635J5035B247H11Ginzburg-Landau Functionalhomogenizationpinning domainvorticesdegree
We consider minimizers of a 2D Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree $d>0$. The pinning term models an unbounded number of small impurities in a superconductor. We prove that for a strongly Type-II superconductor with impurities, the minimizers have exactly $d$ isolated zeros (vortices). These vortices are of degree $1$ and pinned by the impurities. As in the standard case studied by Bethuel, Brezis, and H\'elein in [F. Bethuel, H. Brezis, and F. H\'elein, \textit{Ginzburg-Landau Vortices}, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkh\"auser Boston Inc., Boston, MA, 1994.], the macroscopic location of vortices is governed by vortex/vortex and vortex/boundary repelling effects. In some special cases, we prove that their macroscopic location tends to minimize the renormalized energy of Bethuel-Brezis-H\'elein. In addition, impurities affect the microscopic location of vortices. Our techniques allow us to work with impurities having different sizes. In this situation, we prove that vortices are pinned by the largest impurities.
Indiana University Mathematics Journal
2013
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10.1512/iumj.2013.62.4942
10.1512/iumj.2013.62.4942
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Indiana Univ. Math. J. 62 (2013) 551 - 641
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