  Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$. For $\epsilon>0$
small, we construct non-constant solutions to the Ginzburg-Landau equations
$-\Delta u=\frac{1}{\epsilon^2}(1-|u|^2)u$ in $\Omega$ such that on $\partial
\Omega$ u satisfies $|u|=1$ and $u\wedge \partial_\nu u=0$. These boundary
conditions are called semi-stiff and are intermediate between the Dirichlet and
the homogeneous Neumann boundary conditions. In order to construct such
solutions we use a variational reduction method very similar to the one used by
del Pino-Kowalczyk-Musso to find solutions of the Ginzburg-Landau equations
with Dirichlet and homogeneous Neumann boundary conditions. We obtain the exact
same result as the authors of the aforementioned article obtained for the
Neumann problem. This is because the renormalized energy for the Neumann
problem and for the semi-stiff problem are the same. In particular if $\Omega$
is simply connected a solution with degree one on the boundary always exists
and if $\Omega$ is not simply connected then for any $k\geq 1$ a solution with
$k$ vortices of degree one exists.
