Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, p. 545-580

This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied magnetic field for vortex nucleation is estimated in the London singular limit, and as a by-product, results concerning vortex-pinning and boundary conditions on the interface are obtained.

DOI : https://doi.org/10.1051/cocv/2009009
Classification:  35J60,  35J20,  35J25,  35B40,  35Q55,  82D55
Keywords: generalized Ginzburg-Landau energy functional, proximity effects, global minimizers, unique positive solution, vortices
@article{COCV_2010__16_3_545_0,
     author = {Kachmar, Ayman},
     title = {Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     pages = {545-580},
     doi = {10.1051/cocv/2009009},
     zbl = {1203.35272},
     mrnumber = {2674626},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2010__16_3_545_0}
}
Kachmar, Ayman. Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 545-580. doi : 10.1051/cocv/2009009. http://www.numdam.org/item/COCV_2010__16_3_545_0/

[1] A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity. J. Math. Pures Appl. 80 (2001) 339-372. | Zbl 1027.35123

[2] A. Aftalion, S. Alama and L. Bronsard, Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein condensate. Arch. Rational Mech. Anal. 178 (2005) 247-286. | Zbl 1075.76063

[3] S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions. J. Math. Phys. 46 (2005) 095102. | Zbl 1111.58013

[4] S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains. Comm. Pure Appl. Math. LIX (2006) 0036-0070. | Zbl 1084.82021

[5] N. André, P. Baumann and D. Phillips, Vortex pinning with bounded fields for the Ginzburg-Landau equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 705-729. | Numdam | Zbl 1040.35108

[6] H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II. Comm. Pure Appl. Anal. 8 (2009) 977-998. | Zbl 1180.35495

[7] F. Béthuel and T. Rivière, Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 243-303. | Numdam | Zbl 0842.35119

[8] F. Béthuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Birkhäuser, Boston-Basel-Berlin (1994). | Zbl 0802.35142

[9] S.J. Chapman and G. Richardson, Vortex pinning by inhomogenities in type II superconductors. Phys. D 108 (1997) 397-407. | Zbl 1039.82510

[10] S.J. Chapman, Q. Du and M.D. Gunzburger, A Ginzburg Landau type model of superconducting/normal junctions including Josephson junctions. European J. Appl. Math. 6 (1996) 97-114. | Zbl 0843.35120

[11] P.G. De Gennes, Superconductivity of metals and alloys. Benjamin (1966). | Zbl 0138.22801

[12] Q. Du, M. Gunzburger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Reviews 34 (1992) 529-560. | Zbl 0787.65091

[13] H.J. Fink and W.C.H. Joiner, Surface nucleation and boundary conditions in superconductors. Phys. Rev. Lett. 23 (1969) 120.

[14] T. Giorgi, Superconductors surrounded by normal materials. Proc. Roy. Soc. Edinburgh Sec. A 135 (2005) 331-356. | Zbl 1068.35154

[15] T. Giorgi and D. Phillips, The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model. SIAM J. Math. Anal. 30 (1999) 341-359. | Zbl 0920.35058

[16] J.O. Indekeu, F. Clarysse and E. Montevecchi, Wetting phase transition and superconductivity: The role of suface enhancement of the order parameter in the GL theory. Procceding of the NATO ASI, Albena, Bulgaria (1998).

[17] A. Kachmar, On the ground state energy for a magnetic Schrödinger operator and the effect of the de Gennes boundary condition. J. Math. Phys. 47 (2006) 072106. | Zbl 1112.81035

[18] A. Kachmar, On the perfect superconducting state for a generalized Ginzburg-Landau equation. Asymptot. Anal. 54 (2007) 125-164. | Zbl 1173.35707

[19] A. Kachmar, On the stability of normal states for a generalized Ginzburg-Landau model. Asymptot. Anal. 55 (2007) 145-201. | Zbl 1148.35054

[20] A. Kachmar, Weyl asymptotics for magnetic Schrödinger opertors and de Gennes' boundary condition. Rev. Math. Phys. 20 (2008) 901-932. | Zbl 1167.82024

[21] A. Kachmar, Magnetic Ginzburg-Landau functional with discontinuous constraint. C. R. Math. Acad. Sci. Paris 346 (2008) 297-300. | Zbl 1138.35087

[22] A. Kachmar, Limiting jump conditions for Josephson junctions in Ginzburg-Landau theory. Differential Integral Equations 21 (2008) 95-130. | Zbl pre05844175

[23] L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math. 77 (1999) 1-26. | Zbl 0930.35073

[24] K. Lu and X.-B. Pan, Ginzburg-Landau equation with de Gennes boundary condition. J. Diff. Equ. 129 (1996) 136-165. | Zbl 0873.35088

[25] N.G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | Zbl 0127.31904

[26] E. Montevecchi and J.O. Indekeu, Effects of confinement and surface enhancement on superconductivity. Phys. Rev. B 62 (2000) 661-666.

[27] J. Rubinstein, Six lectures in superconductivity, in Boundaries, Interfaces and Transitions (Banff, AB, 1995), CRM Proc., Lecture Notes 13, Amer. Math. Soc., Providence, RI (1998) 163-184. | Zbl 0921.35161

[28] E. Sandier and S. Serfaty, Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Calc. Var. Partial Differ. Equ. 17 (2003) 17-28. | Zbl 1037.49001

[29] E. Sandier and S. Serfaty, Vortices for the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications 70. Birkhäuser Boston (2007). | Zbl 1112.35002

[30] S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field. I. Commun. Contemp. Math. 1 (1999) 213-254. | Zbl 0944.49007

[31] S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field. II. Commun. Contemp. Math. 1 (1999) 295-333. | Zbl 0964.49005

[32] I.M. Sigal and F. Ting, Pinning of magnetic vortices by an external potential. St. Petresburg Math. J. 16 (2005) 211-236. | Zbl 1067.58015

[33] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures No. 16 (Été, 1965), Les Presses de l'Université de Montréal, Montréal, Québec (1966) 326 p. | Zbl 0151.15501