\(f\)-harmonic maps which map the boundary of the domain to one point in the target. (English) Zbl 1202.58012

Course, Neil

New York J. Math. 13, 423-435 (2007).

Summary: The author considers the class of maps \(u:D\to S^2\), which map the boundary of \(D\) to one point in \(S^2\). 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 \(1/2\int_D f(x) |\nabla u(x)|^2\) – then this need not be true. We shall see that there exist functions \(f:D\to (0,\infty)\) and nonconstant \(f\)-harmonic maps \(u:D\to S^2\) 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.

MSC:

58E20	Harmonic maps, etc.
35J25	Boundary value problems for second-order elliptic equations
53C43	Differential geometric aspects of harmonic maps

Keywords:

harmonic maps; \(f\)-harmonic; boundary; Riemannian surface; constant boundary data