manuscripta mathematica

, Volume 56, Issue 1, pp 1–10

A remark on H1 mappings

  • Robert Hardt
  • Fang -Hua Lin
Article

DOI: 10.1007/BF01171029

Cite this article as:
Hardt, R. & Lin, F.H. Manuscripta Math (1986) 56: 1. doi:10.1007/BF01171029

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.

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Robert Hardt
    • 1
    • 2
  • Fang -Hua Lin
    • 1
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA