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


58E20 Harmonic maps, etc.
49Q99 Manifolds and measure-geometric topics
Full Text: DOI EuDML


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