Advertisement

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

Harmonic Mapping Compact Support Hardy Space Maximal Function Atomic Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations