Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain

Berlyand, Leonid; Golovaty, Dmitry; Rybalko, Volodymyr

doi:10.1016/j.crma.2006.05.013

Calculus of Variations

Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain
[Nonexistence des minimizers de Ginzburg–Landau avec le degré prescrit sur la frontière d'un domaine doublement connexe]

Présenté par : Haïm Brezis

Berlyand, Leonid ¹ ; Golovaty, Dmitry ² ; Rybalko, Volodymyr ³

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
² Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA
³ Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61164 Kharkov, Ukraine

Comptes Rendus. Mathématique, Tome 343 (2006) no. 1, pp. 63-68.

Résumé
Abstract

Soient ω, Ω des ouverts bornés, simplement connexes de $R^{2}$ , et soit $\bar{ω} \subset Ω$ . Dans le domaine annulaire $A = Ω ∖ \bar{ω}$ on considère une classe $J$ des applications à valeurs complexes ayant le module égal à 1 et le degré 1 sur ∂Ω et ∂ω.

On montre que, si $cap (A) < π$ , alors il existe une valeur critique finie $κ_{1}$ du paramètre κ de Ginzburg–Landau, telle que le minimum de l'énergie de Ginzburg–Landau $E_{κ}$ n'est pas atteint dans $J$ pour $κ > κ_{1}$ , tandis qu'il est attaint pour $κ < κ_{1}$ .

Let ω, Ω be bounded simply connected domains in $R^{2}$ , and let $\bar{ω} \subset Ω$ . In the annular domain $A = Ω ∖ \bar{ω}$ we consider the class $J$ of complex valued maps having modulus 1 and degree 1 on ∂Ω and ∂ω.

We prove that, when $cap (A) < π$ , there exists a finite threshold value $κ_{1}$ of the Ginzburg–Landau parameter κ such that the minimum of the Ginzburg–Landau energy $E_{κ}$ not attained in $J$ when $κ > κ_{1}$ while it is attained when $κ < κ_{1}$ .

Reçu le : 2006-01-23
Accepté le : 2006-05-15
Publié le : 2006-06-21

DOI : 10.1016/j.crma.2006.05.013

Affiliations des auteurs :

Berlyand, Leonid ¹ ; Golovaty, Dmitry ² ; Rybalko, Volodymyr ³

@article{CRMATH_2006__343_1_63_0,
     author = {Berlyand, Leonid and Golovaty, Dmitry and Rybalko, Volodymyr},
     title = {Nonexistence of {Ginzburg{\textendash}Landau} minimizers with prescribed degree on the boundary of a doubly connected domain},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--68},
     publisher = {Elsevier},
     volume = {343},
     number = {1},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.05.013/}
}

TY  - JOUR
AU  - Berlyand, Leonid
AU  - Golovaty, Dmitry
AU  - Rybalko, Volodymyr
TI  - Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 63
EP  - 68
VL  - 343
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.05.013/
DO  - 10.1016/j.crma.2006.05.013
LA  - en
ID  - CRMATH_2006__343_1_63_0
ER  -

%0 Journal Article
%A Berlyand, Leonid
%A Golovaty, Dmitry
%A Rybalko, Volodymyr
%T Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain
%J Comptes Rendus. Mathématique
%D 2006
%P 63-68
%V 343
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.05.013/
%R 10.1016/j.crma.2006.05.013
%G en
%F CRMATH_2006__343_1_63_0

Berlyand, Leonid; Golovaty, Dmitry; Rybalko, Volodymyr. Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain. Comptes Rendus. Mathématique, Tome 343 (2006) no. 1, pp. 63-68. doi : 10.1016/j.crma.2006.05.013. http://www.numdam.org/articles/10.1016/j.crma.2006.05.013/

Bibliographie
Cité par

[1] Ahlfors, L. Complex Analysis, McGraw-Hill, 1966

[2] Alama, S.; Bronsard, L. Vortices and pinning effects for the Ginzburg–Landau model in multiply connected domains, Comm. Pure Appl. Math., Volume 59 (2006) no. 1, pp. 36-70

[3] Berlyand, L.; Golovaty, D.; Rybalko, V. Capacity of a multiply-connected domain and nonexistence of Ginzburg–Landau minimizers with prescribed degrees on the boundary, 2006 http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:math/0601018 (Preprint available at) | arXiv

[4] Berlyand, L.; Mironescu, P. Ginzburg–Landau minimizers in perforated domains with prescribed degrees http://desargues.univ-lyon1.fr (Preprint available at)

[5] Berlyand, L.; Mironescu, P. Ginzburg–Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices, J. Funct. Anal. (2006) | DOI

[6] Berlyand, L.; Mironescu, P. Ginzburg–Landau minimizers with prescribed degrees: dependence on domain, C. R. Math. Acad. Sci. Paris, Volume 337 (2003), pp. 375-380

[7] Berlyand, L.V.; Voss, K. Symmetry breaking in annular domains for a Ginzburg–Landau superconductivity model, Proceedings of IUTAM 99/4 Symposium, Sydney, Australia, Kluwer Academic Publishers, 1999, pp. 189-210

[8] Bethuel, F.; Brezis, H.; Hélein, F. Ginzburg–Landau Vortices, Birkhäuser, 2004

[9] Golovaty, D.; Berlyand, L. On uniqueness of vector-valued minimizers of the Ginzburg–Landau functional in annular domains, Calc. Var. Partial Differential Equations, Volume 14 (2002) no. 2, pp. 213-232

[10] Jimbo, S.; Morita, Y. Ginzburg–Landau equations and stable solutions in a rotational domain, SIAM J. Math. Anal., Volume 27 (1996) no. 5, pp. 1360-1385

[11] Rubinstein, J.; Sternberg, P. Homotopy classification of minimizers of the Ginzburg–Landau energy and the existence of permanent currents, Comm. Math. Phys., Volume 179 (1996) no. 1, pp. 257-263

Cité par Sources :