A Ginzburg-Landau model with topologically induced free discontinuities

TL;DR

This paper introduces a variational model combining Ginzburg-Landau and Mumford-Shah features, analyzing vortex clustering with fractional degrees and topologically constrained discontinuities through $ ext{Gamma}$-convergence.

Contribution

It develops a novel model with fractional vortices connected by discontinuities, providing a complete $ ext{Gamma}$-convergence analysis and insights into the structure of minimizers.

Findings

Vortices have fractional degrees $1/m$ with $m \\geq 2$.

Minimizers' discontinuities solve a Steiner-type problem.

Structure of minimizers persists for small $\\varepsilon$.

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 $\Gamma$-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.