The configuration (aj) minimizes the renormalized energy W

  • Fabrice Bethuel
  • Haïm Brezis
  • Frédéric Hélein
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 13)


In Section 1.4 we have introduced the renormalized energy W = W(a,d,g) for a general configuration of points (ai),1≤ i ≤ n, with associated integers d i Z such that \( \sum\nolimits_{{i = 1}}^n {{d_i} = d} \) Throughout this chapter we consider only configurations consisting of exactly d distinct points in G, each one associated to d i =+1.


Unique Minimizer Theorem VIII Harmonic Conjugate Hopf Differential Renormalize Energy 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Fabrice Bethuel
    • 1
  • Haïm Brezis
    • 2
    • 3
  • Frédéric Hélein
    • 4
  1. 1.Laboratoire d’Analyse NumériqueUniversité Paris-SudOrsay CedexFrance
  2. 2.Analyse Numérique Université Pierre et Marie CurieParis Cedex 05France
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA
  4. 4.CMLA, ENS-CachanCachan CedexFrance

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