f-harmonic maps which map the boundary of the domain to one point in the target

摘要

One considers the class of maps u:D → S2, which map the boundary of D to one point in S2. If uwere also harmonic, then it is known that u must be constant. However, if u is insteadf-harmonic -- a critical point of the energy functional 1/2 ∫D f(x) |∇ u(x)|2 --then this need not be true. We shall see that there exist functions f:D → (0,∞) andnonconstant f-harmonic maps u:D → S2 which map the boundary to one point. We will alsosee that there exist nonconstant f for which, there is no nonconstant f-harmonic map in thisclass. Finally, we see that there exists a nonconstant f-harmonic map from the torus to the 2-sphere.