Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model

TL;DR

This paper analyzes the Ginzburg-Landau energy in a 2D superconductor with variable magnetic fields and pinning, deriving asymptotic formulas and discussing critical fields.

Contribution

It provides an accurate asymptotic formula for the energy considering the pinning term and magnetic field variations, which is a novel analysis in this context.

Findings

Derived an asymptotic formula for the minimizing energy.

Demonstrated the influence of the pinning term on energy.

Established asymptotics for the third critical magnetic field.

Abstract

We study the Ginzburg-Landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain $\Omega$. Supposing that the Ginzburg-Landau parameter and the intensity of the magnetic field are large and of the same order, we determine an accurate asymptotic formula for the minimizing energy. This asymptotic formula displays the influence of the pinning term. Also, we discuss the existence of non-trivial solutions and prove some asymptotics of the third critical field.