Variational convergence for functionals of Ginzburg-Landau type

Abstract

In the first part of this paper we prove that functionals of Ginzburg-Landau type for maps from a domain in dimension n+k into R^k converge in a suitable sense to the area functional for surfaces of dimension n (Theorem 1.1). In the second part we modify this result in order to include Dirichlet boundary condition (Theorem 5.5), and, as a corollary, we show that the rescaled energy densities and the Jacobians of minimizers converge to minimal surfaces of dimension n (Corollaries 1.2 and 5.6). Some of these results were announced in the paper "Un risultato di convergenza variazionale per funzionali di tipo Ginzburg-Landau in dimensione qualunque" by the first author

Critical points via Γ-convergence: general theory and applications

“…Basic Γ-convergence results for the Modica-Mortola functional are established in [22,21], and for the Ginzburg-Landau and related functionals in [2], [13]. These automatically yield some results describing asymptotic behavior of minimizing sequences.…”

Mean field limits for Ginzburg-Landau vortices

“…This was proved in the works [Ri,LR1,BBO,JS1,ABO,BBM]. Since the leading order to the energy is not trivial (as opposed to the two-dimensional situation) the dynamics of vortex lines is expected to be driven by it, i.e.…”

“…These are studied in the framework of geometric measure theory, using currents and varifolds, cf. [Ri,JS1,ABO]. The leading order expansion of the energy replacing (2.2) was then shown to be…”

Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

“…with the convention that x 0 ≡ t. In view of the space-times bounds on the Ginzburg-Landau energy, we may invoke the work of Jerrard and Soner [36] (see also [1]), to assert that…”

“…Moreover, it is shown in [36], [1], that 1 π J * can be identified with an integer multiplicity (N -1)-current, whose boundary is exactly J 0 * (the slice at time zero), and…”