  We study the three-dimensional Ginzburg-Landau model of superconductivity for
strong applied magnetic fields varying between the second and third critical
fields. In this regime, it is known from physics that superconductivity should
be essentially restricted to a thin layer along the boundary of the sample.
This leads to the introduction of a Ginzburg-Landau model on a half-space. We
prove that the non-linear Ginzburg-Landau energy on the half-space with
constant magnetic field is a decreasing function of the angle $\nu$ that the
magnetic field makes with the boundary. In the case when the magnetic field is
tangent to the boundary ($\nu=0$), we show that the energy is determined to
leading order by the minimization of a simplified 1D functional in the
direction perpendicular to the boundary. We also study the geometric behavior
of the order parameter near the surface of the sample by constructing formal
solutions with lattice properties.
