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Partial regularity for stationary harmonic maps into spheres. (English) Zbl 0754.58007
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. $$H^{n-2}(\Sigma)=0$$.

##### MSC:
 5.8e+21 Harmonic maps, etc.
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##### References:
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