The Dirichlet problem for harmonic maps from the disk into a sphere.

Abstract

The Dirichlet problem for harmonic maps from the disk into the 2-sphere is a natural, non-linear, generalization of the classical Dirichlet problem. In this context, harmonic maps arise as critical points of the energy functional. For any boundary condition, γ: ∂D to S², the space of extensions of maps to D splits into countably many relative homotopy classes. For boundary values which are rational functions of eⁱᶿ, it is known there are finitely many homotopy classes which contain energy minimizing (and hence harmonic) extensions. It is conjectured that every harmonic extension is a local minimum for the energy, and this view guides much of what follows. This work focuses on the particular case γ(eⁱᶿ) = (cos nθ, sin nθ, 0). Here there are precisely n + 1 relative homotopy classes which contain energy minimizing extensions. The structure of the space of harmonic extensions is studied via the symmetries which act on the space of extensions of γ. When n = 1 or n = 2, this symmetry approach gives an alternative proof to the existence of harmonic extensions in the required classes. Furthermore, by using a symmetry condition, a geometric description of a non-conformal harmonic extension is given. In this setting, there is a continuous S¹-action on the space of extensions. Maps which are invariant under this action are called rotationally invariant. It is shown that every harmonic rotationally invariant extension must be either holomorphic or antiholomorphic. From this result, it is shown that there exist continuous families of harmonic maps in certain homotopy classes. Additional topics which are studied include the finite time blow up of the heat equation for harmonic maps--a geometric description is offered, and two numerical approaches to minimizing the energy in a relative homotopy class: discretizing the heat equation and a direct minimization algorithm.