Harmonic Mappings

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)