  We analyze a mean field model for $3$d anisotropic superconductors with a
layered structure, in the presence of a strong magnetic field. The mean field
model arises as the $Gamma$-limit of the Lawrence-Doniach energy in certain
regimes. A reformulation of the problem based on convex duality allows us to
characterize the first critical field $H_{c_1}$ of the layered superconductor,
up to leading order. In previous work, Alama-Bronsard-Sandier \cite{ABS} have
derived the asymptotic value of $H_{c_1}$ for configurations satisfying
periodic boundary conditions; in that setting describing minimizers of the
Lawrence-Doniach energy reduces to a $2$d problem. In this work, we treat the
physical case without any periodicity assumptions, and are thus led to studying
a delicate and essentially $3$d non-local obstacle problem first derived by
Baldo-Jerrard-Orlandi-Soner \cite{BJOS2} for the isotropic Ginzburg-Landau
energy. We obtain a characterization of $H_{c_1}$ using the special anisotropic
structure of the mean field model.
