One considers the class of maps u:D → S², which map the boundary of D to one point in S². If u were also harmonic, then it is known that u must be constant. However, if u is instead f-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,∞) and nonconstant f-harmonic maps u:D → S² which map the boundary to one point. We will also see that there exist nonconstant f for which, there is no nonconstant f-harmonic map in this class. Finally, we see that there exists a nonconstant f-harmonic map from the torus to the 2-sphere.

Research supported by Swiss National Science Foundation grant number 200020-107652/1 and EPSRC award number 00801877.