A Ginzburg–Landau model with topologically induced free discontinuities

Goldman, Michael; Merlet, Benoit; Millot, Vincent

doi:10.5802/aif.3388

A Ginzburg–Landau model with topologically induced free discontinuities
[Un modèle de type Ginzburg–Landau avec des discontinuités libres topologiquement induites]

Goldman, Michael ¹ ; Merlet, Benoit ² ; Millot, Vincent ³

¹ Laboratoire Jacques-Louis Lions (CNRS, UMR 7598) Université Paris Diderot 75005 Paris (France)
² Univ. Lille, CNRS, UMR 8524 Inria – Laboratoire Paul Painlevé 59000 Lille (France)
³ LAMA, Univ Paris Est Creteil, Univ Gustave Eiffel, UPEM, CNRS 94010 Créteil (France)

Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2583-2675.

Résumé
Abstract

Nous étudions un modèle variationnel en deux dimensions qui combine les caractéristiques des fonctionnelles de Ginzburg–Landau et de Mumford–Shah. Comme dans la théorie classique de Ginzburg–Landau (et dans le régime de faible énergie) un nombre prescrit de vortex apparaît ; le modèle autorise aussi la formation de lignes de discontinuité dont l’énergie pénalise la longueur. Le phénomène nouveau est que les vortex ont un degé fractionnaire $1 / m$ prescrit et qu’ils doivent être connectés par les lignes de discontinuité pour former des agrégats de degré total entier. Vortex et discontinuités sont donc couplés par une contrainte topologique. Comme dans le modèle de Ginzburg–Landau, l’énergie contient une échelle de longueur $ε > 0$ . Nous faisons une analyse complète de la $Γ -$ convergence de ce modèle lorsque $ε ↓ 0$ dans le régime de faible énergie. Nous étudions ensuite la structure des minimiseurs du problème limite et montrons en particulier que les lignes de saut d’un tel minimiseur sont solutions d’une variante du problème de Steiner. Enfin, nous établissons que pour $ε > 0$ petit, les minimiseurs du problème initial possèdent la même structure, du moins loin des vortex.

We study a variational model which combines features of the Ginzburg–Landau model in 2D and of the Mumford–Shah functional. As in the classical Ginzburg–Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1 / m$ with $m \geq 2$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg–Landau model, the energy is parameterized by a small length scale $ε > 0$ . We perform a complete $Γ$ -convergence analysis of the model as $ε ↓ 0$ in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small $ε > 0$ , the minimizers of the original problem have the same structure away from the limiting vortices.

Reçu le : 2019-01-21
Révisé le : 2019-09-20
Accepté le : 2019-12-20
Publié le : 2021-04-15

DOI : 10.5802/aif.3388

Classification : 35Q56, 49S05, 82D55, 49J10, 49S05
Keywords: Free discontinuities, Ginzburg–Landau, Steiner problem, Calculus of Variations
Mot clés : Problèmes de discontinuités libres, Ginzburg–Landau, Problème de Steiner, Calcul des Variations

Affiliations des auteurs :

Goldman, Michael ¹ ; Merlet, Benoit ² ; Millot, Vincent ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A {Ginzburg{\textendash}Landau} model with topologically induced free discontinuities},
     journal = {Annales de l'Institut Fourier},
     pages = {2583--2675},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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 Goldman, Michael; Merlet, Benoit; Millot, Vincent. A Ginzburg–Landau model with topologically induced free discontinuities. Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2583-2675. doi : 10.5802/aif.3388. https://aif.centre-mersenne.org/articles/10.5802/aif.3388/

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