March  2012, 7(1): 179-196. doi: 10.3934/nhm.2012.7.179

Renormalized Ginzburg-Landau energy and location of near boundary vortices

1. 

Department of Mathematics, Penn State University, University Park, State College, PA 16802, United States

2. 

Mathematical Division, B. I. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 61103, Ukraine

3. 

Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907

Received  April 2011 Revised  December 2011 Published  February 2012

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. Networks and Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179
References:
[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.

show all references

References:
[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.

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