Estimates of the energy of type-II superconductor (Q1863660)

The authors consider the following Ginzburg-Landau free energy of a type-II superconductor \[ J(u,A)=\frac{1}{2} \int_\Omega |\nabla_A u|^2 +|h-h_{ex}|^2 +\frac{1}{2\varepsilon^2} (a(x)-|u|^2)^2, \] where \(u\) is the order parameter, \(\kappa=1/\varepsilon\) is the Ginzburg-Landau parameter and \(h_{ex}\) is a prescribed magnetic field. In contrast to the original form of the functional where \(a\equiv 1\), \(a\) is a function whose minimas correspond to material impurities. The main result of the paper is an asymptotic expression for the free energy in the mixed phase, in which normal and superconducting phase can coexist. More precisely, they prove \[ J(u_\varepsilon,A_\varepsilon) \thicksim \frac{1}{2} \left(\int_\Omega a(x) dx) h_{ex} \ln \frac{1}{\varepsilon\sqrt{h_{ex}}}(1+ o(1)\right) \quad \text{as } \varepsilon \to 0. \]