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

TL;DR

This paper derives a new explicit formula for the microscopic renormalized energy in a pinned Ginzburg-Landau model with impurities, providing insights into the minimal energy configurations and their dependence on impurity shape.

Contribution

It introduces a sharp decomposition of the minimal energy for a Dirichlet functional with an L-infinity weight, leading to an explicit expression for circular impurities.

Findings

Explicit formula for microscopic renormalized energy with circular impurity

Decomposition of minimal energy for weighted Dirichlet functional

Analysis of energy minimization in specific impurity configurations

Abstract

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.