  We consider minimizing harmonic maps $u$ from $\Omega \subset \mathbb{R}^n$
into a closed Riemannian manifold $\mathcal{N}$ and prove:
  (1) an extension to $n \geq 4$ of Almgren and Lieb's linear law. That is, if
the fundamental group of the target manifold $\mathcal{N}$ is finite, we have
\[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial \Omega} |\nabla_T
u|^{n-1} \,d \mathcal{H}^{n-1}; \]
  (2) an extension of Hardt and Lin's stability theorem. Namely, assuming that
the target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singular
set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary
data.
  In dimension $n=3$ both results are shown to hold with weaker hypotheses,
i.e., only assuming that the trace of our map lies in the fractional space
$W^{s,p}$ with $s \in (\frac{1}{2},1]$ and $p \in [2,\infty)$ satisfying $sp
\geq 2$. We also discuss sharpness.
