English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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, ifthe fundamental group of the target manifold $\mathcal{N}$ is finite, we have\[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial \Omega} |\nabla_Tu|^{n-1} \,d \mathcal{H}^{n-1}; \]  (2) an extension of Hardt and Lin's stability theorem. Namely, assuming thatthe target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singularset of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundarydata.  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.

On the size of the singular set of minimizing harmonic maps