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Traveling fronts guided by the environment for reaction-diffusion equations
Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes
1. | Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States |
2. | Mathematical Division, B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103 Kharkiv, Ukraine |
References:
[1] |
A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl. (9), 80 (2001), 339-372.
doi: 10.1016/S0021-7824(00)01180-6. |
[2] |
S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions, J. Math. Phys., 46 (2005), 39 pp.
doi: 10.1063/1.2010354. |
[3] |
S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., 59 (2006), 36-70.
doi: 10.1002/cpa.20086. |
[4] |
H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998.
doi: 10.3934/cpaa.2009.8.977. |
[5] |
E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995. |
[6] |
G. R. Berdiyorov, M. V. Milosević and F. M. Peeters, Novel commensurability effects in superconducting films with antidot arrays, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.207001. |
[7] |
M. Dos Santos and O. Misiats, Ginzburg-Landau model with small pinning domains, Netw. Heterog. Media, 6 (2011), 715-753.
doi: 10.3934/nhm.2011.6.715. |
[8] |
M. Dos Santos, P. Mironescu and O. Misiats, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I : The zero degree case, Comm. Contemp. Math., 13 (2011), 885-914.
doi: 10.1142/S021919971100449X. |
[9] |
M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case, preprint. |
[10] |
I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974. |
[11] |
A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint, ESAIM Control Optim. Calc. Var., 16 (2010), 545-580.
doi: 10.1051/cocv/2009009. |
[12] |
L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26.
doi: 10.1007/BF02791255. |
[13] |
P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997.
doi: 10.1007/978-3-0348-8886-8. |
[14] |
E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Annales IHP, Analyse Non Linéaire, 17 (2000), 119-145.
doi: 10.1016/S0294-1449(99)00106-7. |
[15] |
M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188.
doi: 10.1007/BFb0084935. |
show all references
References:
[1] |
A. Aftalion, E. Sandier and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl. (9), 80 (2001), 339-372.
doi: 10.1016/S0021-7824(00)01180-6. |
[2] |
S. Alama and L. Bronsard, Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions, J. Math. Phys., 46 (2005), 39 pp.
doi: 10.1063/1.2010354. |
[3] |
S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., 59 (2006), 36-70.
doi: 10.1002/cpa.20086. |
[4] |
H. Aydi and A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II, Commun. Pure Appl. Anal., 8 (2009), 977-998.
doi: 10.3934/cpaa.2009.8.977. |
[5] |
E. J. Balder, "Lectures on Young Measures," Cah. de Ceremade, 1995. |
[6] |
G. R. Berdiyorov, M. V. Milosević and F. M. Peeters, Novel commensurability effects in superconducting films with antidot arrays, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.207001. |
[7] |
M. Dos Santos and O. Misiats, Ginzburg-Landau model with small pinning domains, Netw. Heterog. Media, 6 (2011), 715-753.
doi: 10.3934/nhm.2011.6.715. |
[8] |
M. Dos Santos, P. Mironescu and O. Misiats, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part I : The zero degree case, Comm. Contemp. Math., 13 (2011), 885-914.
doi: 10.1142/S021919971100449X. |
[9] |
M. Dos Santos, The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term. Part II: The non-zero degree case, preprint. |
[10] |
I. Ekeland and R. Temam, "Analyse Convexe et Problemes Variationnels," (French) Collection Etudes Mathematiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974. |
[11] |
A. Kachmar, Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint, ESAIM Control Optim. Calc. Var., 16 (2010), 545-580.
doi: 10.1051/cocv/2009009. |
[12] |
L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77 (1999), 1-26.
doi: 10.1007/BF02791255. |
[13] |
P. Pedregal, "Parametrized Measures and Variational Principles," Birkhauser, 1997.
doi: 10.1007/978-3-0348-8886-8. |
[14] |
E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Annales IHP, Analyse Non Linéaire, 17 (2000), 119-145.
doi: 10.1016/S0294-1449(99)00106-7. |
[15] |
M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes Math., Springer, (1990), 152-188.
doi: 10.1007/BFb0084935. |
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