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Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

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Abstract

We study the regularity of the minimizer u λ for the functional F λ(u,f)=∫Ω|∇u|2 + λ∫Ω|u−f{2 over all maps u∈H 1(Ω, S 2). We prove that for some suitable functions f every minimizer u λ is smooth in Ω if λ ⩽ λ0 and for the same functions f, u λ has singularities when λ is large enough.

Résumé

On étudie la régularité des minimiseurs u λ du problème de minimisation minueH 1(Ω,S2)(∫Ω|∇u|2 + λ∫Ω|u−f{2. On montre que pour certaines fonctions f, u λ est régulière lorsque λ ⩽ λ0 et pour les mêmes f, si λ est assez grand, alors u λ possède des singularités.

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Authors and Affiliations

  1. Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris cedex 05, France

    R. Hadiji & F. Zhou

  2. Department of Mathematics, University of Wuhan, 430072, Wuhan, China

    F. Zhou

Authors
  1. R. Hadiji
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  2. F. Zhou
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Hadiji, R., Zhou, F. Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon. Potential Anal 1, 385–400 (1992). https://doi.org/10.1007/BF00301791

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  • DOI: https://doi.org/10.1007/BF00301791

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