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Communications in Mathematical Physics

, Volume 107, Issue 4, pp 649–705 | Cite as

Harmonic maps with defects

  • Haïm Brezis
  • Jean-Michel Coron
  • Elliott H. Lieb
Article

Abstract

Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation. Extensions of these problems are also solved, e.g. points are replaced by “holes,” ℝ3,S2 is replaced by ℝ N ,SN−1 or by ℝ N , ℝPN−1, the latter being appropriate for the theory of liquid crystals.

Keywords

Neural Network Statistical Physic Complex System Liquid Crystal Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Haïm Brezis
    • 1
  • Jean-Michel Coron
    • 2
  • Elliott H. Lieb
    • 3
  1. 1.Département de MathématiquesUniversité P. et M. CurieParis Cedex 05France
  2. 2.Département de MathématiquesEcole PolytechniquePalaiseau CedexFrance
  3. 3.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

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