# Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 545-580
- ISSN: 1292-8119

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topKachmar, Ayman. "Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 545-580. <http://eudml.org/doc/250815>.

@article{Kachmar2010,

abstract = {
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.
},

author = {Kachmar, Ayman},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Generalized Ginzburg-Landau energy
functional; proximity effects; global minimizers; unique positive
solution; vortices; generalized Ginzburg-Landau energy functional; unique positive solution},

language = {eng},

month = {7},

number = {3},

pages = {545-580},

publisher = {EDP Sciences},

title = {Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint},

url = {http://eudml.org/doc/250815},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Kachmar, Ayman

TI - Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 545

EP - 580

AB -
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.

LA - eng

KW - Generalized Ginzburg-Landau energy
functional; proximity effects; global minimizers; unique positive
solution; vortices; generalized Ginzburg-Landau energy functional; unique positive solution

UR - http://eudml.org/doc/250815

ER -

## References

top- A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity. J. Math. Pures Appl.80 (2001) 339–372.
- 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.
- S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions. J. Math. Phys.46 (2005) 095102.
- 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.
- N. André, P. Baumann and D. Phillips, Vortex pinning with bounded fields for the Ginzburg-Landau equation. Ann. Inst. H. Poincaré Anal. Non Linéaire20 (2003) 705–729.
- 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.
- F. Béthuel and T. Rivière, Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire12 (1995) 243–303.
- F. Béthuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Birkhäuser, Boston-Basel-Berlin (1994).
- S.J. Chapman and G. Richardson, Vortex pinning by inhomogenities in type II superconductors. Phys. D108 (1997) 397–407.
- 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.
- P.G. de Gennes, Superconductivity of metals and alloys. Benjamin (1966).
- Q. Du, M. Gunzburger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Reviews34 (1992) 529–560.
- H.J. Fink and W.C.H. Joiner, Surface nucleation and boundary conditions in superconductors. Phys. Rev. Lett.23 (1969) 120.
- T. Giorgi, Superconductors surrounded by normal materials. Proc. Roy. Soc. Edinburgh Sec. A135 (2005) 331–356.
- 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.
- 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).
- 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.
- A. Kachmar, On the perfect superconducting state for a generalized Ginzburg-Landau equation. Asymptot. Anal.54 (2007) 125–164.
- A. Kachmar, On the stability of normal states for a generalized Ginzburg-Landau model. Asymptot. Anal.55 (2007) 145–201.
- A. Kachmar, Weyl asymptotics for magnetic Schrödinger opertors and de Gennes' boundary condition. Rev. Math. Phys.20 (2008) 901–932.
- A. Kachmar, Magnetic Ginzburg-Landau functional with discontinuous constraint. C. R. Math. Acad. Sci. Paris346 (2008) 297–300.
- A. Kachmar, Limiting jump conditions for Josephson junctions in Ginzburg-Landau theory. Differential Integral Equations21 (2008) 95–130.
- L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math.77 (1999) 1–26.
- K. Lu and X.-B. Pan, Ginzburg-Landau equation with de Gennes boundary condition. J. Diff. Equ.129 (1996) 136-165.
- N.G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic equations. Ann. Sc. Norm. Sup. Pisa17 (1963) 189–206.
- E. Montevecchi and J.O. Indekeu, Effects of confinement and surface enhancement on superconductivity. Phys. Rev. B62 (2000) 661–666.
- J. Rubinstein, Six lectures in superconductivity, in Boundaries, Interfaces and Transitions (Banff, AB, 1995), CRM Proc., Lecture Notes13, Amer. Math. Soc., Providence, RI (1998) 163–184.
- 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.
- E. Sandier and S. Serfaty, Vortices for the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications70. Birkhäuser Boston (2007).
- S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field. I. Commun. Contemp. Math.1 (1999) 213–254.
- S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field. II. Commun. Contemp. Math.1 (1999) 295–333.
- I.M. Sigal and F. Ting, Pinning of magnetic vortices by an external potential. St. Petresburg Math. J.16 (2005) 211–236.
- 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.

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