Sine-Laplace equation,sink - Laplace equation and harmonic maps

The relationship between harmonic maps from R² to S², H², S^T,1, S^1,1(–1) and the ± sinh — Laplace, ± sine — Laplace equation is found respectively. Existence theorems of some boundary value problems for the above harmonic maps are obtained. In the cases of H², S^1,1(+1), S^1,1(–1) the results are global.Research supported partially by the Institute for Applied Mathematics, Sonderforschungsbereich 72 of the University of Bonn