A Ginzburg-Landau model with topologically induced free discontinuities

Michael Goldman (1) , Benoît Merlet (2) , Vincent Millot (1)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 RAPSODI - Reliable numerical approximations of dissipative systems

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.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Michael Goldman, Benoît Merlet, Vincent Millot. A Ginzburg-Landau model with topologically induced free discontinuities. Annales de l'Institut Fourier, 2020, Université de Grenoble. Annales de l'Institut Fourier, 70 (6), pp.2583--2675. ⟨hal-01643795⟩