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Harmonic maps with defects

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Abstract

Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS 2 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,S 2 is replaced by ℝN,S N−1 or by ℝN, ℝP N−1, the latter being appropriate for the theory of liquid crystals.

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Author notes

  1. Elliott H. Lieb

    Present address: Departments of Mathematics and Physics, Princeton University, Jadwin Hall, P.O. Box 708, 08544, Princeton, NJ, USA

Authors and Affiliations

  1. Département de Mathématiques, Université P. et M. Curie, 4 pl. Jussieu, F-75252, Paris Cedex 05, France

    Haïm Brezis

  2. Département de Mathématiques, Ecole Polytechnique, F-91128, Palaiseau Cedex, France

    Jean-Michel Coron

  3. Institut des Hautes Etudes Scientifiques, F-91440, Bures-sur-Yvette, France

    Elliott H. Lieb

Authors
  1. Haïm Brezis
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  2. Jean-Michel Coron
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  3. Elliott H. Lieb
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Additional information

Communicated by A. Jaffe

Work partially supported by U.S. National Science Foundation grant PHY 85-15288-A02

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Brezis, H., Coron, JM. & Lieb, E.H. Harmonic maps with defects. Commun.Math. Phys. 107, 649–705 (1986). https://doi.org/10.1007/BF01205490

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