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

Leonid Berlyand; Dmitry Golovaty; Volodymyr Rybalko

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

Presented by: Haïm Brezis

Leonid Berlyand ¹; Dmitry Golovaty ²; Volodymyr Rybalko 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, Volume 343 (2006) no. 1, pp. 63-68.

Abstract
Résumé

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 $κ < κ_{1}$ .

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 $κ < κ_{1}$ .

Received: 2006-01-23
Accepted: 2006-05-15
Published online: 2006-06-21

DOI: 10.1016/j.crma.2006.05.013

Author's affiliations:

Leonid Berlyand ¹; Dmitry Golovaty ²; Volodymyr Rybalko 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

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     author = {Leonid Berlyand and Dmitry Golovaty and Volodymyr Rybalko},
     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},
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     number = {1},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.013},
     language = {en},
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Leonid Berlyand; Dmitry Golovaty; Volodymyr Rybalko. Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain. Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 63-68. doi : 10.1016/j.crma.2006.05.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.013/

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