A remark on H¹ mappings

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.