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From the Ginzburg-Landau model to vortex lattice problems. (English) Zbl 1252.35034

The authors are interested in deriving a “Coulombian renormalized energy” from the Ginzburg-Landau model of superconductivity. One of the key tools in the arguments is the associated renormalized energy combined with an abstract method for lower bounds for two-scale energies using ergodic theory. The main results establish various properties, such as the existence of minimizers. In particular, by means of some results in number theory, it is established that among lattice configurations the triangular lattice is the unique minimizer.

MSC:

35B25 Singular perturbations in context of PDEs
82D55 Statistical mechanics of superconductors
35J20 Variational methods for second-order elliptic equations
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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[1] Alberti G., Choksi R., Otto F.: Uniform Energy Distribution for an Isoperimetric Problem With Long-range Interactions. J. Amer. Math. Soc. 22(2), 569–605 (2009) · Zbl 1206.49046 · doi:10.1090/S0894-0347-08-00622-X
[2] Alberti G., Müller S.: A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54(7), 761–825 (2001) · Zbl 1021.49012 · doi:10.1002/cpa.1013
[3] Aftalion A., Serfaty S.: Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c_2. Selecta Math 13(2), 183–202 (2007) · Zbl 1138.82034 · doi:10.1007/s00029-007-0043-7
[4] Aydi, H.: Doctoral Dissertation, Université Paris-12, 2004
[5] Aydi H., Sandier E.: Vortex analysis of the periodic Ginzburg-Landau model. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1223–1236 (2009) · Zbl 1171.35480 · doi:10.1016/j.anihpc.2008.09.004
[6] Becker M.E.: Multiparameter groups of measure-preserving transformations: a simple proof of Wiener ergodic theorem. Ann. Probability 9(3), 504–509 (1981) · Zbl 0468.28020 · doi:10.1214/aop/1176994423
[7] Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau Vortices. Basel-Boston: Birkhäuser, 1994
[8] Billingsley, P.: Convergence of probability measures. Second edition. New York: Wiley & Sons, 1999 · Zbl 0944.60003
[9] Borodin, A., Serfaty, S.: Renormalized Energy Concentration in Random Matrices. Preprint, http://arxiv.org/abs/1201.2853v2 [math.PR], 2012 · Zbl 1276.60007
[10] Brezis H.: Problèmes unilatéraux. J. Math. Pures Appl. (9) 51, 1–168 (1972)
[11] Brezis, H., Kinderlehrer, D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1973/74)
[12] Brezis H., Serfaty S.: A variational formulation for the two-sided obstacle problem with measure data. Commun. Contemp. Math. 4(2), 357–374 (2002) · Zbl 1146.35372 · doi:10.1142/S0219199702000671
[13] Caffarelli L.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3-4), 155–184 (1977) · Zbl 0386.35046 · doi:10.1007/BF02392236
[14] Caffarelli L.A., Friedman A.: Convexity of solutions of semilinear elliptic equations. Duke Math. J. 52(2), 431–456 (1985) · Zbl 0599.35065 · doi:10.1215/S0012-7094-85-05221-4
[15] Cassels J.W.S.: On a problem of Rankin about the Epstein zeta-function. Proc. Glasgow Math. Assoc. 4, 73–80 (1959) · Zbl 0103.27602 · doi:10.1017/S2040618500033906
[16] Chen X., Oshita Y.: An application of the modular function in nonlocal variational problems. Arch. Rat. Mech. Anal. 186, 109–132 (2007) · Zbl 1147.74024 · doi:10.1007/s00205-007-0050-z
[17] Cohn, D.L.: Measure Theory. Basel-Boston: Birkhaüser, 1980 · Zbl 0436.28001
[18] Cohn H., Kumar A.: Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20(1), 99–148 (2007) · Zbl 1198.52009 · doi:10.1090/S0894-0347-06-00546-7
[19] DeGennes, P.G.: Superconductivity of metal and alloys. New York-Amsterdam: Benjamin, 1966
[20] Diananda P.H.: Notes on two lemmas concerning the Epstein zeta-function. Proc. Glasgow Math. Assoc. 6, 202–204 (1964) · Zbl 0128.04501 · doi:10.1017/S2040618500035036
[21] Dolbeault J., Monneau R.: Convexity Estimates for Nonlinear Elliptic Equations and Application to Free Boundary Problems. Ann. Inst. H. Poincaré, Analyse Non Linéaire 19(6), 903–926 (2002) · Zbl 1034.35047 · doi:10.1016/S0294-1449(02)00106-3
[22] Dudley, R.: Real analysis and probability. Cambridge Studies in Advanced Mathematics, 74, Cambridge: Cambridge University Press, 2002 · Zbl 1023.60001
[23] Ennola V.: A remark about the Epstein zeta function. Proc. Glasg. Math. Assoc. 6, 198–201 (1964) · Zbl 0128.04402 · doi:10.1017/S2040618500035024
[24] Ennola V.: On a problem about the Epstein zeta-function. Proc. Cambridge Philos. Soc. 60, 855–875 (1964) · Zbl 0146.05504 · doi:10.1017/S0305004100038330
[25] Fournais S., Helffer B.: On the third critical field in Ginzburg-Landau theory. Comm. Math. Phys. 266, 153–196 (2006) · Zbl 1107.58009 · doi:10.1007/s00220-006-0006-4
[26] Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. In: Progress in Nonlinear Differential Equations, vol. 77, Berlin-Heidelberg-Newyork: Springer, 2010 · Zbl 1256.35001
[27] Frehse J.: On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital. (4) 6, 312–315 (1972) · Zbl 0261.49021
[28] Friedman A., Phillips D.: The free boundary of a semilinear elliptic equation. Trans. Amer. Math. Soc. 282(1), 153–182 (1984) · Zbl 0552.35079 · doi:10.1090/S0002-9947-1984-0728708-4
[29] Goldman,D., Muratov, C., Serfaty, S.: The {\(\Gamma\)}-limit of the two dimensional Ohta-Kawasaki energy. Part I: Droplet density. http://arxiv.org/abs/1201.0222v1 [math.ph], 2012 · Zbl 1296.82018
[30] Goldman, D., Muratov, C., Serfaty, S.: The {\(\Gamma\)}-limit of the two dimensional Ohta-Kawasaki energy. Part II: Derivation of the renormalized energy. In preparation · Zbl 1305.35134
[31] Hervé R. M., Hervé M.: Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire 11(4), 427–440 (1994)
[32] Isakov, V.M.: Inverse theorems on the smoothness of potentials. Differencial’nye Uravnenija 11, 66–74, 202 (1975) · Zbl 0309.31012
[33] Jerrard R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999) · Zbl 0928.35045 · doi:10.1137/S0036141097300581
[34] Jerrard R.L., Soner H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations 14(2), 151–191 (2002) · Zbl 1034.35025
[35] Kawohl B.: When are solutions to nonlinear elliptic boundary value problems convex?. Comm. PDE 10, 1213–1225 (1985) · Zbl 0587.35026 · doi:10.1080/03605308508820404
[36] Kinderlehrer D., Nirenberg L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2), 373–391 (1977) · Zbl 0352.35023
[37] Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics, 88. London-Newyork: Academic Press, 1980 · Zbl 0457.35001
[38] Korevaar N., Lewis J.L.: Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Rat. Mech. Anal. 97(1), 19–32 (1987) · Zbl 0624.35031 · doi:10.1007/BF00279844
[39] Lang, S.: Elliptic functions. New York: Springer-Verlag, 1987 · Zbl 0615.14018
[40] Lang, S.: Introduction to Arakelov theory. Springer-Verlag, New York, 1988 · Zbl 0667.14001
[41] Mironescu P.: Les minimiseurs locaux pour l’équation de Ginzburg-Landau sont à symétrie radiale. C. R. Acad. Sci. Paris Sér. I Math. 323(6), 593–598 (1996) · Zbl 0858.35038
[42] Monneau, R.: A brief overview on the obstacle problem. European Congress of Mathematics, Vol. II, Progr. Math., 202, Basel: Birkhäuser, 2001, pp 303–312 · Zbl 1027.35164
[43] Montgomery H.L.: Minimal Theta functions. Glasgow Math J. 30(1), 75–85 (1988) · Zbl 0639.10017 · doi:10.1017/S0017089500007047
[44] Muratov C.: Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299, 45–87 (2010) · Zbl 1205.82107 · doi:10.1007/s00220-010-1094-8
[45] Radin C.: The ground state for soft disks. J. Stat. Phys. 26(2), 367–372 (1981) · doi:10.1007/BF01013177
[46] Rankin R.A.: A minimum problem for the Epstein zeta function. Proc. Glasgow Math. Assoc 1, 149–158 (1953) · Zbl 0052.28005 · doi:10.1017/S2040618500035668
[47] Rivière N.M.: Singular integrals and multiplier operators. Ark. Mat. 9, 243–278 (1971) · Zbl 0244.42024 · doi:10.1007/BF02383650
[48] Saff, E., Totik, V.: Logarithmic potentials with external fields. Berlin: Springer-Verlag, 1997 · Zbl 0881.31001
[49] Saint-James, D., Sarma, G., Thomas, E.J.: Type-II superconductivity. Oxford: Pergamon Press, 1969
[50] Sandier E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152(2), 379–403 (1998) · Zbl 0908.58004 · doi:10.1006/jfan.1997.3170
[51] Sandier E., Serfaty S.: Global Minimizers for the Ginzburg-Landau Functional below the First critical Magnetic Field. Annales Inst. H. Poincaré, Anal. non linéaire 17(1), 119–145 (2000) · Zbl 0947.49004 · doi:10.1016/S0294-1449(99)00106-7
[52] Sandier E., Serfaty S.: The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model. SIAM J. Math. Anal. 34(4), 939–956 (2003) · Zbl 1030.82015 · doi:10.1137/S0036141002406084
[53] Sandier E., Serfaty S.: Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement. Analysis and PDE 4-5, 757–795 (2011) · Zbl 1270.35150 · doi:10.2140/apde.2011.4.757
[54] Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model. Basel-Boston: Birkhäuser, 2007 · Zbl 1112.35002
[55] Erratum to [SS4], pp. 148–151, available at http://www.ann.jussieu.fr/\(\sim\)serfaty/publis.html
[56] Sandier, E., Serfaty, S.: 2D Coulomb Gases and the Renormalized Energy, Preprint, http://arxiv.org/abs/1201.3503v1 [math.ph], 2012 · Zbl 1328.82006
[57] Serfaty, S., Tice, I.: Lorentz Space Estimates for the Coulombian Renormalized Energy, to appear in Comm. Contemp. Math, available at http://arxiv.org/abs/1105.3960v1 [math-ph], 2012 · Zbl 1255.35061
[58] Struwe M.: On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions. Diff. and Int. Eqs. 7(5-6), 1613–1624 (1994) · Zbl 0809.35031
[59] Theil F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006) · Zbl 1113.82016 · doi:10.1007/s00220-005-1458-7
[60] Tinkham, M.: Introduction to superconductivity. Second edition. New York: McGraw-Hill, 1996
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