On the size of the singular set of minimizing harmonic maps

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 Liebs 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} | abla_T u|^{n-1} ,d mathcal{H}^{n-1}; ] (2) an extension of Hardt and Lins 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.