Partial regularity for stationary harmonic maps into spheres (Q1189742)

Let \(\Omega\subset \mathbb{R}^ n\) denote an open set and consider a weak solution \(u\in H^{1,2}(\Omega,S^{m-1})\) of \(-\Delta u=u\cdot |\nabla u|^ 2\), i.e. a weak solution of the Euler-Lagrange equation of the energy functional \(\int_ \Omega|\nabla u|^ 2dx\) among all mappings from \(\Omega\) into the sphere \(S^{m-1}\). If in addition the monotonicity inequality holds for \(u\) then \(u\in C^ \infty(\Omega-\Sigma,\mathbb{R}^ m)\) for some relatively closed subset \(\Sigma\) of \(\Omega\) s.t. \(H^{n-2}(\Sigma)=0\).