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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

##### MSC:
 58E20 Harmonic maps, etc. 53C20 Global Riemannian geometry, including pinching
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##### References:
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