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 is done by obtaining a sharp decomposition for the minimal energy of a Dirichlet type functional with an L $\infty$-weight. In particular we get an explicit expression of the microscopic renormalized energy for a circular impurity. We proceed also to the minimization of this renormalized energy in some cases.