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A remark on H1 mappings

Authors:	Robert Hardt Fang -Hua Lin

Institution:	(1) School of Mathematics, University of Minnesota, 55455 Minneapolis, MN, USA;(2) Courant Institute of Mathematical Sciences, New York University, 10012 New York, NY, USA

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 over U(g) is strictly smaller than the infimum of over the continuous functions in U(g). There are some generalizations to higher dimensions.Research partially supported by the National Science FoundationResearch supported by an Alfred P. Sloan Graduate Fellowship

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