Analysis of Minimizers of the Lawrence-Doniach Energy for Superconductors in Applied Fields

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder, $ \Omega\times[0,L]$, in $\mathbb{R}^3$, where $\Omega$ is a bounded simply connected Lipschitz domain in $\mathbb{R}^2$. For an applied magnetic field $\vec{H}_{ex}=h_{ex}\vec{e}_{3}$ that is perpendicular to the layers with $\left|\ln\epsilon\right|\ll h_{ex}\ll\epsilon^{-2}$ as $\epsilon\rightarrow 0$, where $\epsilon$ is the reciprocal of the Ginzburg-Landau parameter, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as $\epsilon$ and the interlayer distance $s$ tend to zero. Under appropriate assumptions on $s$ versus $\epsilon$, we establish comparison results between the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropic Ginzburg-Landau energy. As a consequence, our asymptotic formula also describes the minimum three-dimensional anisotropic energy as $\epsilon$ tends to zero.