Title :

A Ginzburg-Landau model with topologically induced free discontinuities

Author(s) :

Goldman, Michael [Auteur]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Merlet, Benoît [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Millot, Vincent [Auteur]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]

Journal title :

Annales de l'Institut Fourier
Université de Grenoble. Annales de l'Institut Fourier

Pages :

2583--2675

Publisher :

Association des Annales de l'Institut Fourier

Publication date :

2020-04-15

ISSN :

0373-0956

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Physique mathématique [math-ph]

English abstract : [en]

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 ...
Show more >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≥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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Théorie géométrique de la mesure et applications
Analyse des singularités topologiques dans quelques problèmes issus de la physique mathématique
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL