Calculus of Variations and Geometric Measure Theory
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M. Goldman - B. Merlet - V. Millot

A Ginzburg-Landau model with topologically induced free discontinuities

created by goldman on 22 Nov 2017
modified on 24 Nov 2017

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Submitted Paper

Inserted: 22 nov 2017
Last Updated: 24 nov 2017

Year: 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 $m\geq2$ 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 $\Gamma$-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.


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