1. A Ginzburg-Landau model with topologically induced free discontinuities
- Author
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Goldman, Michael, Merlet, Benoît, Millot, Vincent, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), We thank R. Badal, M. Cicalese, L. De Luca, and M. Ponsiglione for telling us about their result [BCLP16] and giving us an early access to a preprint version. We also thank M. Dos Santos for pointing out the paper [ILR00]. The authors have been supported by the Agence Nationale de la Recherche through the grants ANR-12-BS01-0014-01 (Geometrya), ANR-14-CE25-0009-01 (MAToS), and by the PGMO research project COCA. BM was partially supported by the INRIA team RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01)., ANR-12-BS01-0014,GEOMETRYA,Théorie géométrique de la mesure et applications(2012), ANR-14-CE25-0009,MAToS,Analyse des singularités topologiques dans quelques problèmes issus de la physique mathématique(2014), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe
- Subjects
Mathematics - Analysis of PDEs ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Condensed Matter::Superconductivity ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Analysis of PDEs (math.AP) - Abstract
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 $\varepsilon>0$. We perform a complete $��$-convergence analysis of the model as $\varepsilon\downarrow0$ 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 $\varepsilon>0$, the minimizers of the original problem have the same structure away from the limiting vortices.
- Published
- 2017
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