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.
Categories: math.AP math.DG