Harmonic maps with defects

Haïm Brezis, Jean Michel Coron, Elliott H. Lieb

Research output: Contribution to journalArticlepeer-review

303 Scopus citations

Abstract

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

Original languageEnglish (US)
Pages (from-to)649-705
Number of pages57
JournalCommunications In Mathematical Physics
Volume107
Issue number4
DOIs
StatePublished - Dec 1 1986

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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