## A remark on $$H^ 1$$ mappings.(English)Zbl 0618.58015

Authors’ summary: ”With $${\mathbb{B}}=\{x\in {\mathbb{R}}^ 3:| x| <1\}$$, we here construct, for each positive integer N, a smooth function $$g: \partial {\mathbb{B}}\to {\mathbb{S}}^ 2$$ of degree zero so that there must be at least N singular points for any map that minimizes the energy $${\mathcal E}(u)=\int_{{\mathbb{B}}}| \nabla u|^ 2dx$$ in the family $$U(g)=\{u\in H^ 1({\mathbb{B}},{\mathbb{S}}^ 2):$$ $$u| \partial {\mathbb{B}}=g\}$$. The infimum of $${\mathcal E}$$ over U(g) is strictly smaller than the infimum of $${\mathcal E}$$ over the continuous functions in U(g). There are some generalizations to higher dimensions.”
Reviewer: N.Jacob

### MSC:

 58E20 Harmonic maps, etc. 49Q99 Manifolds and measure-geometric topics
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### References:

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