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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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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DOI: https://doi.org/10.1007/BF01205490