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

Hardt, Robert; Lin, Fang-Hua

Manuscr. Math. 56, 1-10 (1986).

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

Keywords:

H\({}^ 1\)-maps; energy minimizing maps with singularities

References:

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