Boundary Regularity and the Dirichlet Problem for Harmonic Maps

In a previous paper [10] we developed an interior regularity theory for energy minimizing harmonic maps into Riemannian manifolds. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed Dirichlet boundary condition. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds, the boundary and the data. The reader may refer to Theorem 2.7 for a statement of the precise result. It is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior. This is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres S+~ which map the boundary S~~ = 9S+" to a point (I <j'<n — 2), and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior. Many authors have worked on boundary regularity for this general type of problem. We mention Hildebrandt and Widman [5] and Hamilton [4] as having obtained important results specifically for harmonic maps. Morrey had obtained the boundary regularity for domain dimension n = 2 in conjunction with his investigation of the Plateau problem in Riemannian manifolds [8]. In §3 of this paper, we observe that the direct method gives solvability of the Dirichlet problem under reasonable hypotheses on the manifolds. We give, as an application, an amusing proof of a theorem of Sacks and Uhlenbeck [9] on the existence of minimal 2-spheres representing the second homotopy group of a manifold. The same method gives smooth harmonic representations for πk(N) for a certain class of manifolds N. These are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal (see Proposition 3.4). In the last section of the paper we discuss approximation of L\ maps by smooth maps. We give a simple example of an L\ map from the three-dimensional ball to the two-sphere which is not an L\ limit of continuous maps. We