Harmonic Mappings

  • Alain Bensoussan
  • Jens Frehse
Part of the Applied Mathematical Sciences book series (AMS, volume 151)

Abstract

We treat here only a small aspect of this large field. In particular, we do not consider harmonic mappings between manifolds, but only from a domain Ω of R n into the unit sphere of R N . This is, however, sufficient to cover many of the analytical difficulties. There is a large litterature, starting with the famous paper of J. Eells, J.H. Sampson [22]. Let us introduce the problem: Let Ω be a bounded open subset of R n , n ≥ 2, and
$$ g:\bar \Omega \to {R^N},\;Lipschitz,\;\left| {g\left( x \right)} \right| = 1\;\forall x. $$
(5.1)
Find u such that
$$ \begin{gathered} u \in {H^1}\left( {\Omega ;{R^N}} \right)\;\left| u \right| = 1,{\left. u \right|_{\partial \Omega }} = g, \hfill \\ u\;\min imizes\;\int_\Omega {{{\left| {Du} \right|}^2}dx} . \hfill \\ \end{gathered} $$
(5.2)

Keywords

Manifold Convolution 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alain Bensoussan
    • 1
  • Jens Frehse
    • 2
  1. 1.CNESParisFrance
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany

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