Partial Differential Equations
Ginzburg–Landau minimizers with prescribed degrees: dependence on domain
[Minimiseurs de l'énergie de Ginzburg–Landau avec degrés prescrits : dépendance du domaine]

Présenté par : Haïm Brezis

Berlyand, Leonid1 ; Mironescu, Petru2
1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2 Département de mathématiques, Université Paris-Sud, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Tome 337 (2003) no. 6, pp. 375-380.

On étudie des minimiseurs de l'énergie de Ginzburg–Landau dans un domaine annulaire à trous. Les conditions aux limites sont des degrés prescrits : degrés 1 et −1 sur le bord du domaine annulaire, degré 0 sur les bords des trous. En fonction de la H1-capacité du domaine, les minimiseurs ont deux types de comportement, qualitativement différents. On décrit aussi le comportement des minimiseurs quand le paramètre de Ginzburg–Landau tend vers ∞.

We study minimizers of the Ginzburg–Landau functional in an annular type domain with holes. We assume degrees 1 and −1 on the boundary of the annulus, degree 0 on the boundaries of the holes. Two types of qualitatively different behavior of minimizers occur, depending on the value of the H1-capacity of the domain. We also describe the asymptotic behavior of minimizers as the coherency length tends to ∞.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00331-5
Berlyand, Leonid 1 ; Mironescu, Petru 2

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2 Département de mathématiques, Université Paris-Sud, 91405 Orsay cedex, France 
@article{CRMATH_2003__337_6_375_0,
     author = {Berlyand, Leonid and Mironescu, Petru},
     title = {Ginzburg{\textendash}Landau minimizers with prescribed degrees: dependence on domain},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--380},
     publisher = {Elsevier},
     volume = {337},
     number = {6},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00331-5},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00331-5/}
}
TY  - JOUR
AU  - Berlyand, Leonid
AU  - Mironescu, Petru
TI  - Ginzburg–Landau minimizers with prescribed degrees: dependence on domain
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 375
EP  - 380
VL  - 337
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00331-5/
DO  - 10.1016/S1631-073X(03)00331-5
LA  - en
ID  - CRMATH_2003__337_6_375_0
ER  - 
%0 Journal Article
%A Berlyand, Leonid
%A Mironescu, Petru
%T Ginzburg–Landau minimizers with prescribed degrees: dependence on domain
%J Comptes Rendus. Mathématique
%D 2003
%P 375-380
%V 337
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00331-5/
%R 10.1016/S1631-073X(03)00331-5
%G en
%F CRMATH_2003__337_6_375_0
Berlyand, Leonid; Mironescu, Petru. Ginzburg–Landau minimizers with prescribed degrees: dependence on domain. Comptes Rendus. Mathématique, Tome 337 (2003) no. 6, pp. 375-380. doi : 10.1016/S1631-073X(03)00331-5. http://www.numdam.org/articles/10.1016/S1631-073X(03)00331-5/

[1] Berlyand, L. Effective properties of superconducting and superfluid composites, Int. J. Modern Phys. B, Volume 12 (1998), pp. 3063-3073

[2] Berlyand, L.; Khruslov, E. Homogenization of harmonic maps and superconducting composites, SIAM J. Appl. Math., Volume 59 (1999), pp. 1892-1916

[3] L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers with prescribed degrees: emergence of vortices, dependence on domain and on coherency length, in press

[4] Berlyand, L.; Voss, K. Symmetry breaking in annular domains for a Ginzburg–Landau superconductivity model, Proceedings of IUTAM 99/4 Symposium (Sydney, Australia), Kluwer Academic, 1999

[5] Bethuel, F.; Brezis, H.; Coleman, B.D.; Hélein, F. Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Rational Mech. Anal., Volume 118 (1992), pp. 149-168

[6] Bethuel, F.; Brezis, H.; Hélein, F. Asymptotics for the minimization of a Ginzburg–Landau functional, Calc. Var., Volume 1 (1993), pp. 123-148

[7] Bethuel, F.; Brezis, H.; Hélein, F. Ginzburg–Landau Vortices, Birkhäuser, Boston, MA, 1994

[8] Brezis, H.; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983), pp. 437-477

[9] Boutet de Monvel-Berthier, A.; Georgescu, V.; Purice, R. A boundary value problem related to the Ginzburg–Landau problem, Comm. Math. Phys., Volume 142 (1991), pp. 1-23

[10] Donnelly, R.J.; Fetter, A.L. Stability of superfluid flow in an annulus, Phys. Rev. Lett., Volume 17 (1966), pp. 747-750

[11] Golovaty, D.; Berlyand, L. On uniqueness of vector-valued minimizers of the Ginzburg–Landau functional in annular domains, Calc. Var., Volume 14 (2002), pp. 213-232

[12] Jimbo, S.; Morita, Y. Ginzburg–Landau equations and stable solutions in a rotational domain, SIAM J. Math. Anal., Volume 27 (1996), pp. 1360-1385

[13] Lassoued, L.; Mironescu, P. Ginzburg–Landau type energy with discontinuous constraint, J. d'Analyse, Volume 77 (1999), pp. 1-26

[14] Maz'ya, V. Sobolev Spaces, Springer-Verlag, 1985

[15] Mironescu, P. Explicit bounds for solutions to a Ginzburg–Landau type equation, Rev. Roumaine Math. Pures Appl., Volume 41 (1996), pp. 263-271

Cité par Sources :