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