Feuilleter
no. 5
Towards an asymptotic analysis of the anisotropic Ginzburg–Landau model
[Vers une analyse asymptotique de la fonctionnelle de Ginzburg–Landau anisotrope]
Dmitry Golovaty1 ; Petru Mironescu2 ; Peter Sternberg3
1 Department of Mathematics, University of Akron, Akron, OH 44325-4002, USA
2 Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France
3 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1367-1432

We develop a set of tools for the asymptotic analysis of minimizers of the anisotropic Ginzburg–Landau energy functional among the admissible competitors with Dirichlet boundary datum of negative degree $-D$. As a byproduct of our analysis, we prove that the energy of a minimizer is $K\ln (1/\varepsilon )+o(\ln (1/\varepsilon ))$, where $K$ depends only on $D$ and the material constants that enter into the expression for the energy.

Nous proposons plusieurs outils pour l’analyse des minimiseurs de la fonctionnelle de Ginzburg–Landau anisotrope avec donnée de Dirichlet de degré négatif $-D$. Comme une application de notre analyse, nous montrons que l’énergie des minimiseurs est $K\ln (1/\varepsilon )+o(\ln (1/\varepsilon ))$, où la constante $K$ dépend uniquement de $D$ et des constantes physiques qui apparaissent dans l’expression de l’énergie.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1836
Classification : 35B25, 35B40, 35J20
Keywords: Anisotropic Ginzburg–Landau, Oseen–Frank, nematic, vortex, Pohozaev identity

Dmitry Golovaty 1 ; Petru Mironescu 2 ; Peter Sternberg 3

1 Department of Mathematics, University of Akron, Akron, OH 44325-4002, USA
2 Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France
3 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AFST_2025_6_34_5_1367_0,
     author = {Dmitry Golovaty and Petru Mironescu and Peter Sternberg},
     title = {Towards an asymptotic analysis of the anisotropic {Ginzburg{\textendash}Landau} model},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1367--1432},
     year = {2025},
     publisher = {Universit\'e de Toulouse, Toulouse},
     volume = {Ser. 6, 34},
     number = {5},
     doi = {10.5802/afst.1836},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1836/}
}
TY  - JOUR
AU  - Dmitry Golovaty
AU  - Petru Mironescu
AU  - Peter Sternberg
TI  - Towards an asymptotic analysis of the anisotropic Ginzburg–Landau model
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2025
SP  - 1367
EP  - 1432
VL  - 34
IS  - 5
PB  - Université de Toulouse, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1836/
DO  - 10.5802/afst.1836
LA  - en
ID  - AFST_2025_6_34_5_1367_0
ER  - 
%0 Journal Article
%A Dmitry Golovaty
%A Petru Mironescu
%A Peter Sternberg
%T Towards an asymptotic analysis of the anisotropic Ginzburg–Landau model
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2025
%P 1367-1432
%V 34
%N 5
%I Université de Toulouse, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1836/
%R 10.5802/afst.1836
%G en
%F AFST_2025_6_34_5_1367_0
Dmitry Golovaty; Petru Mironescu; Peter Sternberg. Towards an asymptotic analysis of the anisotropic Ginzburg–Landau model. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1367-1432. doi: 10.5802/afst.1836

[1] Shmuel Agmon; Avron Douglis; Louis Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., Volume 12 (1959), pp. 623-727 | Zbl | DOI

[2] Fabrice Bethuel; Haïm Brézis; Frédéric Hélein Ginzburg–Landau vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser, 1994 | Zbl | DOI

[3] Haïm Brézis; Frank Merle; Tristan Rivière Quantization effects for -Δu=u(1-|u| 2 ) in ℝ 2 , Arch. Ration. Mech. Anal., Volume 126 (1994) no. 1, pp. 35-58 | DOI | Zbl | MR

[4] Sergio Campanato Equazioni ellittiche del IIdeg ordine espazi ℒ (2,λ) , Ann. Mat. Pura Appl. (4), Volume 69 (1965), pp. 321-381 | DOI | Zbl | MR

[5] Sean Colbert-Kelly; Daniel Phillips Analysis of a Ginzburg–Landau type energy model for smectic C * liquid crystals with defects, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 6, pp. 1009-1026 | DOI | Zbl | Numdam

[6] Avron Douglis; Louis Nirenberg Interior estimates for elliptic systems of partial differential equations, Commun. Pure Appl. Math., Volume 8 (1955), pp. 503-538 | DOI | Zbl | MR

[7] Mariano Giaquinta Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, 105, Princeton University Press, 1983 | DOI | Zbl | MR

[8] David Gilbarg; Neil S. Trudinger Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, 224, Springer, 1983 | Zbl

[9] Dmitry Golovaty; Peter Sternberg; Raghavendra Venkatraman A Ginzburg–Landau-type problem for highly anisotropic nematic liquid crystals, SIAM J. Math. Anal., Volume 51 (2019) no. 1, pp. 276-320 | DOI | Zbl | MR

[10] Zheng-Chao Han; Itai Shafrir Lower bounds for the energy of S 1 -valued maps in perforated domains, J. Anal. Math., Volume 66 (1995), pp. 295-305 | DOI | Zbl | MR

[11] Robert L. Jerrard Lower Bounds for Generalized Ginzburg–Landau Functionals, SIAM J. Math. Anal., Volume 30 (1999) no. 4, pp. 721-746 | DOI | MR | Zbl

[12] Michał Kowalczyk; Xavier Lamy; Panayotis Smyrnelis Entire vortex solutions of negative degree for the anisotropic Ginzburg–Landau system, Arch. Ration. Mech. Anal., Volume 245 (2022) no. 1, pp. 565-586 | DOI | Zbl | MR

[13] Petru Mironescu Les minimiseurs locaux pour l’équation de Ginzburg–Landau sont à symétrie radiale, C. R. Math., Volume 323 (1996) no. 6, pp. 593-598 | Zbl

[14] Charles B. jun. Morrey On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior, Am. J. Math., Volume 80 (1958), pp. 198-218 | DOI | Zbl | MR

[15] Ivan G. Petrowsky Sur l’analyticité des solutions des systèmes d’équations différentielles, Rec. Math. Moscou, n. Ser., Volume 5 (1939), pp. 3-70 | Zbl | MR

[16] Manuel del Pino; Patricio L. Felmer On the basic concentration estimate for the Ginzburg–Landau equation, Differ. Integral Equ., Volume 11 (1998) no. 5, pp. 771-779 | Zbl | MR

[17] Étienne Sandier Locally minimising solutions of -Δu=u(1-|u| 2 ) in R 2 , Proc. R. Soc. Edinb., Sect. A, Math., Volume 128 (1998) no. 2, pp. 349-358 | DOI | Zbl | MR

[18] Étienne Sandier Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., Volume 152 (1998) no. 2, pp. 379-403 | DOI | Zbl

[19] Étienne Sandier; Sylvia Serfaty Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser, 2007 | Zbl | MR | DOI

[20] Michael Struwe On the asymptotic behavior of minimizers of the Ginzburg–Landau model in 2 dimensions, Differ. Integral Equ., Volume 7 (1994) no. 5-6, pp. 1613-1624 | Zbl

[21] Michael Struwe Erratum: “On the asymptotic behavior of minimizers of the Ginzburg–Landau model in 2 dimensions”, Differ. Integral Equ., Volume 8 (1995) no. 1, p. 224 | Zbl | MR

[22] Epifanio G. Virga Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation, 8, Chapman & Hall, 1994 | Zbl | MR | DOI

Cité par Sources :