Large solutions for harmonic maps in two dimensions
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 \textit{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.
scientific article; zbMATH DE number 3844673