Renormalized Ginzburg-Landau energy and location of near boundary vortices

  • 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.

    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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  • 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.


    [1] N. André and I. Shafrir, On the minimizers of a Ginzburg-Landau-type energy when the boundary condition has zeros, Adv. Differential Equations, 9 (2004), 891-960.
    [2] P. Bauman, N. N. Carlson and D. Phillips, On the zeros of solutions to Ginzburg-Landau type systems, SIAM J. Math. Anal., 24 (1993), 1283-1293. doi: 10.1137/0524073
    [3] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of Variations and PDEs, 1 (1993), 123-148. doi: 10.1007/BF01191614
    [4] F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices," Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994.
    [5] A. Boutet de Monvel-Berthier, V. Georgescu and R. Purice, A boundary value problem related to the Ginzburg-Landau model, Comm. Math. Phys., 142 (1991), 1-23. doi: 10.1007/BF02099170
    [6] L. Berlyand and P. Mironescu, Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices, J. Funct. Anal., 239 (2006), 76-99. doi: 10.1016/j.jfa.2006.03.006
    [7] L. Berlyand and V. Rybalko, Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc., 12 (2010), 1497-1531. doi: 10.4171/JEMS/239
    [8] L. Berlyand and K. Voss, Symmetry breaking in annular domains for a Ginzburg-Landau superconductivity model, in "Proceedings of IUTAM 99/4 Symposium," Sydney, Australia, Kluwer Acad. Publ., (2001), 189-200.
    [9] R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal., 30 (1999), 721-746. doi: 10.1137/S0036141097300581
    [10] M. Kurzke, Boundary vortices in thin magnetic films, Calc. Var. Partial Differential Equations, 26 (2006), 1-28.
    [11] P. Mironescu, Explicit bounds for solutions to a Ginzburg-Landau type equations, Rev. Roumain Math. Pure Appl., 41 (1996), 263-271.
    [12] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
    [13] B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math., 160 (1988), 1-17. doi: 10.1007/BF02392271
  • This article has been cited by:

    1. Rémy Rodiac, Paúl Ubillús, Renormalized energies for unit-valued harmonic maps in multiply connected domains, 2022, 128, 18758576, 413, 10.3233/ASY-211712
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