Explicit expression of the microscopic renormalized energy for a pinned Ginzburg–Landau functional

We get a new expression of the microscopic renormalized energy for a pinned Ginzburg–Landau type energy modeling small impurities. This renormalized energy occurs in the simplified 2D Ginzburg–Landau model ignoring the magnetic field as well as the full planar magnetic model. As in the homogenous case, when dealing with heterogeneities, the notion of renormalized energies is crucial in the study of the variational Ginzburg–Landau type problems. The key point of this article is the location of singularities inside a small impurity. The microscopic renormalized energy is defined via the minimization of a Dirichlet type functional with an L ∞ -weight. Namely, the main result of the present article is a sharp asymptotic estimate for the minimization of a weighted Dirichlet energy evaluated among S 1 -valued maps defined on a perforated domain with shrinking holes (in the spirit of the famous work of Bethuel–Brezis–Hélein). The renormalized energy depends on the center of the holes and it is expressed in a computable way. In particular we get an explicit expression of the microscopic renormalized energy when the weight in the Dirichlet energy models an impurity which is a disk. In this case we proceed also to the minimization of the renormalized energy.