A Ginzburg–Landau model with topologically induced free discontinuities
Michael D. Goldman, Benoît Merlet, Vincent Millot · Annales de l’institut Fourier · 2021
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.