Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors with Josephson constant $ \lambda $ and Ginzburg-Landau parameter $ 1/\epsilon $ in a bounded generalized cylinder $ D = \Omega\times[0, L] $ in $ \mathbb{R}^3 $, where $ \Omega $ is a bounded simply connected Lipschitz domain in $ \mathbb{R}^2 $. Our main result is that in an applied magnetic field $ \vec{H}_{ex} = h_{ex}\vec{e}_{3} $ which is perpendicular to the layers with $ \left|\ln\epsilon\right|\ll h_{ex}\ll\epsilon^{-2} $, the minimum Lawrence-Doniach energy is given by $ \frac{|D|}{2}h_{ex}\ln\frac{1}{\epsilon\sqrt{h_{ex}}}(1+o_{\epsilon, s}(1)) $ as $ \epsilon $ and the interlayer distance $ s $ tend to zero. We also prove estimates on the behavior of the order parameters, induced magnetic field, and vorticity in this regime. Finally, we observe that as a consequence of our results, the same asymptotic formula holds for the minimum anisotropic three-dimensional Ginzburg-Landau energy in $ D $ with anisotropic parameter $ \lambda $ and $ o_{\epsilon, s}(1) $ replaced by $ o_{\epsilon}(1) $.