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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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[1] Aubin, Th.: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl.55, 269-296 (1976) · Zbl 0336.53033
[2] Brezis, H., Coron, J. M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. (to appear)
[3] Brezis, H. Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. (to appear)
[4] Giaquinta, M., Hildebrandt, S.: A priori estimates for harmonic mappings. J. Reine Angew. Math.336, 124-164 (1982) · Zbl 0508.58015
[5] Jost, J.: The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values. Invent. Math. (to appear) · Zbl 0551.58012
[6] Lemaire, L.: Applications harmoniques de surfaces riemanniennes. J. Diff. Geom.13, 51-78 (1978) · Zbl 0388.58003
[7] Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. (to appear) · Zbl 0527.42011
[8] Lions, P. L.: The concentration-compactness principle in the Calculus of Variations; The limit case. (to appear) · Zbl 0704.49006
[9] Nirenberg, L.: Topics in nonlinear functional analysis, New York University Lecture Notes 1973-1974
[10] Schoen, R., Uhlenbeck, K.: Boundary regularity and miscellaneous results on harmonic maps. J. Diff. Geom. (to appear) · Zbl 0547.58020
[11] Taubes, C.: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on ?3. Commun. Math. Phys.86, 257-298, 299-320 (1982) · Zbl 0514.58016
[12] Wente, H.: The Dirichlet problem with a volume constraint, Manuscripta Math.11, 141-157 (1974) · Zbl 0268.35031
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