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A Ginzburg-Landau model with topologically induced free discontinuities
- Source :
- Annales de l'Institut Fourier, Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2020, Université de Grenoble. Annales de l'Institut Fourier, 70 (6), pp.2583--2675, Annales de l'Institut Fourier, 2020, Université de Grenoble. Annales de l'Institut Fourier, 70 (6), pp.2583--2675
- Publication Year :
- 2017
- Publisher :
- arXiv, 2017.
-
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.
Details
- ISSN :
- 03730956 and 17775310
- Database :
- OpenAIRE
- Journal :
- Annales de l'Institut Fourier, Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2020, Université de Grenoble. Annales de l'Institut Fourier, 70 (6), pp.2583--2675, Annales de l'Institut Fourier, 2020, Université de Grenoble. Annales de l'Institut Fourier, 70 (6), pp.2583--2675
- Accession number :
- edsair.doi.dedup.....62f5f6fb7beef898808a0e58d421985d
- Full Text :
- https://doi.org/10.48550/arxiv.1711.08668