Large solutions for harmonic maps in two dimensions. (English) Zbl 0532.58006

The authors establish the existence of local (but not absolute) minimum of the energy functional \(E\) for the set \({\mathcal E}\) of maps \(u\in H^ 1(\Omega;\mathbb S^ 2)\) defined on the unit disk \(\Omega \subset {\mathbb R}^ 2\) into the Euclidean 2-sphere \(S^ 2\) (or a Riemannian surface homeomorphic to \(S^ 2)\) satisfying a nonconstant boundary condition \(u=\gamma\) on \(\partial \Omega\). (Nonexistence of nontrivial \(u\) for \(\gamma =\text{const}\) has been shown by L. Lemaire [J. Differ. Geom. 13, 51–78 (1979; Zbl 0388.58003)].
The proof is based on considering the functional \(Q(u)=\frac{1}{4\pi}\int_{\Omega}u\cdot u_ x\wedge u_ y\) and showing that at least one of the two infima \(I_{\pm}=\text{Inf}\{E(u)\mid u\in {\mathcal E},\quad Q(u)-Q(\underline u)=\pm 1\}\) is achieved, where \(\underline u\in {\mathcal E}\) such that \(E(\underline u)=\text{Inf}\{E(u)\mid u\in {\mathcal E}\}.\) A detailed study of the case when \(\gamma\) is a small circle on \(S^ 2\) is also given.
Reviewer: G. Tóth


58E20 Harmonic maps, etc.
53C20 Global Riemannian geometry, including pinching
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