Abstract
We examine the existence problem for harmonic maps between the three-dimensional ball and the two-sphere. We utilize results on the classification of harmonic maps into hemispheres and a result on the regularity of the weak limit of energy minimizers over the class of axially symmetric maps to establish the existence of asmooth harmonic extension for boundary data suitably “concentrated” away from the axis of symmetry. In addition, we establish convergence results for the harmonic map heat flow problem for suitably “concentrated” axially symmetric initial and boundary data.
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This work was completed while the author was supported by the Sonderforschungsbereich 256 at the Universität Bonn, and he would like to thank the faculty and staff of the SFB 256. In addition, thanks go to Professor Robert Hardt for several helpful discussions on this material. Part of this work follows on from a section of the author’s thesis [G2], and he wishes to thank his advisor, Professor FangHua Lin.