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

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 ΩΩ. 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.