  We consider $\mathbb{S}^2$-valued maps on a domain
$\Omega\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy
with vertical penalization in $\Omega$ and horizontal penalization on
$\partial\Omega$. We first show the global minimality of universal constant
configurations in a specific range of the physical parameters using a
Poincar\'e-type inequality. Then, we prove that any energy minimizer takes its
values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce
uniqueness of minimizers up to the action of the appropriate symmetry group. We
also prove a comparison principle for minimizers with different penalizations.
Finally, we apply these results to a problem on a ball and show radial symmetry
and monotonicity of minimizers. In dimension $N=2$ our results can be applied
to the Oseen--Frank energy for nematic liquid crystals and micromagnetic energy
in a thin-film regime.
