  We study the asymptotic behavior, when $\varepsilon\to0$, of the minimizers
$\{u_\varepsilon\}_{\varepsilon>0}$ for the energy \begin{equation*}
E_\varepsilon(u)=\int_{\Omega}\Big(|\nabla
u|^2+\big(\frac{1}{\varepsilon^2}-1\big)|\nabla|u||^2\Big), \end{equation*}
over the class of maps $u\in H^1(\Omega,{\mathbb R}^2)$ satisfying the boundary
condition $u=g$ on $\partial\Omega$, where $\Omega$ is a smooth, bounded and
simply connected domain in ${\mathbb R}^2$ and $g:\partial\Omega\to S^1$ is a
smooth boundary data of degree $D\ge1$. The motivation comes from a simplified
version of the Ericksen model for nematic liquid crystals with variable degree
of orientation. We prove convergence (up to a subsequence) of
$\{u_\varepsilon\}$ towards a singular $S^1$-valued harmonic map $u_*$, a
result that resembles the one obtained in \cite{BBH} for an analogous problem
for the Ginzburg-Landau energy. There are however two striking differences
between our result and the one involving the Ginzburg-Landau energy. First, in
our problem the singular limit $u_*$ may have singularities of degree strictly
larger than one. Second, we find that the principle of \enquote{equi-partition}
holds for the energy of the minimizers, i.e., the contributions of the two
terms in $E_\varepsilon(u_\varepsilon)$ are essentially equal.
