Abstract
With\(\mathbb{B} = \left\{ {\varepsilon \mathbb{R}^3 :\left| x \right|< 1} \right\}\), 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\(\varepsilon \left( u \right) = \int\limits_\mathbb{B} {\left| {\nabla u} \right|} ^2 dx\) in the family\(U\left( g \right) : \left\{ {u\varepsilon H^1 \left( {\mathbb{B},\mathbb{S}^2 } \right) : u\left| {\partial \mathbb{B} = g} \right|} \right\}\). The infimum of ε over U(g) is strictly smaller than the infimum of ε over the continuous functions in U(g). There are some generalizations to higher dimensions.
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Research partially supported by the National Science Foundation
Research supported by an Alfred P. Sloan Graduate Fellowship
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Hardt, R., Lin, F.H. A remark on H1 mappings. Manuscripta Math 56, 1–10 (1986). https://doi.org/10.1007/BF01171029
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DOI: https://doi.org/10.1007/BF01171029