Line-energy Ginzburg-Landau models : zero-energy states
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 187-202.

We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful analysis of the corresponding weak solutions by the method of characteristics.

Classification : 35B65,  35J60,  35L65,  74G65,  82D30
@article{ASNSP_2002_5_1_1_187_0,
author = {Jabin, Pierre-Emmanuel and Otto, Felix and Perthame, Beno{\^\i}t},
title = {Line-energy Ginzburg-Landau models : zero-energy states},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {187--202},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {1},
year = {2002},
zbl = {1072.35051},
mrnumber = {1994807},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/}
}
Jabin, Pierre-Emmanuel; Otto, Felix; Perthame, BenoÎt. Line-energy Ginzburg-Landau models : zero-energy states. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 187-202. http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/

[1] L. Ambrosio - C. De Lellis - C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations 9 (1999), 327-355. | MR 1731470 | Zbl 0960.49013

[2] P. Aviles - Y. Giga, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for grasient fields, Proc. Roy. Soc. Edinburgh 129A (1999), 1-17. | MR 1669225 | Zbl 0923.49008

[3] F. Béthuel - H. Brézis - F. Hélein, “Ginzburg-Landau vortices”, Progress in Nonlinear Differential Equations and their Applications, Birkhauser, 1994. | MR 1269538 | Zbl 0802.35142

[4] M. Cessenat, Théorèmes de trace ${L}^{p}$ pour les espaces de fonctions de la neutronique, C.R. Acad. Sci. Paris Sér. I 299 (1984), 834 and 300 (1985), 89. | MR 777741 | Zbl 0568.46030

[5] A. Desimone - R. V. Kohn - S. Müller - F. Otto, A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edinburgh 131 (2001), 833-844. | MR 1854999 | Zbl 0986.49009

[6] A. Desimone - R. V. Kohn - S. Müller - F. Otto, Magnetic microstructures, a paradigm of multiscale problems, Proceedings of ICIAM, to appear. | Zbl 0991.82038

[7] H. Federer, “Geometric measure theory”, Springer-Verlag, 1969. | MR 257325 | Zbl 0176.00801

[8] R. Howard - A. Treibergs, A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mountain J. Math. 25 (1995), n. 2, 635-684. | MR 1336555 | Zbl 0909.53002

[9] P. E. Jabin - B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging, Comm. Pure Appl. Math. 54 (2001), 1096-1109. | MR 1835383 | Zbl 1124.35312

[10] W. Jin - R. V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci 10 (2000), 355-390. | MR 1752602 | Zbl 0973.49009

[11] T. Rivière - S. Serfaty, Limiting domain wall energy in micromagnetism, Comm. Pure Appl. Math. 54 (2001), 294-338. | MR 1809740 | Zbl 1031.35142

[12] T. Rivière - S. Serfaty, Compactness, kinetic formulation, and entropies for a problem related to micromagnetics, preprint (2001). | MR 1974456 | Zbl 1094.35125

[13] S. Ukai, Solutions of the Boltzmann equation, In: “Pattern and waves”, North-Holland 1986. | MR 882376 | Zbl 0633.76078

[14] A. Vasseur, Strong traces for solutions to multidimensional scalar conservation laws, Arch. Rational Mech. Anal., to appear. | MR 1869441 | Zbl 0999.35018