The blow-up of \(p\)-harmonic maps (Q1315078)

Theorem: Let \(\Omega\) denote an open bounded set in \(\mathbb{R}^ k\); if \(u_ n \rightharpoonup u\) weakly in \(H^{1,p} (\Omega,\mathbb{R}^ k)\) and \(\lim \partial_ \alpha (| \nabla u_ n|^{p-2} \partial_ \alpha u_ n)=0\) in the sense of distributions, then \(\partial_ \alpha (|\nabla u|^{p-2} \partial_ \alpha u)=0\). This has applications in the partial regularity of \(p\)-stationary maps \(\Omega\to S^{k-1}\).