A $\Gamma$-convergence result for 2D type-I superconductors

Abstract

We consider a 2D non-standard Modica–Mortola-type functional. This functional arises from the Ginzburg–Landau theory of type-I superconductors in the case of an infinitely long sample and in the regime of comparable penetration and coherence lengths. We prove that the functional $\Gamma$-converges to the perimeter functional. This result is a first step in understanding how to extend the results of Conti, Otto, Goldman, and Serfaty (2018) to the regime of non-vanishing Ginzburg–Landau parameter $\kappa$.