Dirichlet Problems for Harmonic Maps from Regular Domains
University of Copenhagen, Department of Mathematics Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. E-mail: fuglede{at}math.ku.dk
Received 4 November 2003. Revision received 10 May 2004. Revision received 28 August 2004.
Given an open set 
of compact closure in Rm, the classical Dirichlet problem is to extend a given continuous function 
: 
R to a continuous function 
such that 
is harmonic (that is, satisfies the Laplace equation) in . The set is termed regular if the Dirichlet problem has a (necessarily unique) solution for any continuous boundary function . For example, every simply connected planar domain is regular (but may have a 'bad' boundary , for instance, a fractal).
In this article it is shown that, when is regular (in the above sense), every continuous map from to a simply connected complete Riemannian manifold (N, h) of sectional curvature at most 0 has a unique continuous extension 
which is harmonic in . This is done with Rm replaced more generally by an m-dimensional Riemannian manifold (M, g).
The proof relies on the unique solvability of the corresponding variational Dirichlet problem (for any open set 
M). And for that, the above target manifold N can be replaced more generally by any simply connected complete geodesic space Y of curvature at most 0 in the sense of A. D. Alexandrov. Assuming that M satisfies the Poincaré inequality, we show that, for any map : M Y of finite energy in the sense of N. J. Korevaar and R. M. Schoen, there exists a unique map : M Y with = on 
such that minimizes the energy of all maps M Y which agree with on . If is regular then is continuous at any point of at which is continuous. For a Lipschitz (and hence regular) domain M, existence and uniqueness of the variational solution was obtained by Korevaar and Schoen, and earlier for suitable polyhedral targets Y by Gromov and Schoen. Instead of the Riemannian manifold M our domain space can still more generally be an admissible Riemannian polyhedron (as studied in the recent Cambridge Tract by J. Eells and the present author); the variational solution is then in general only Hölder continuous in . The proofs of the stated results of this article rely in part on potential theory relative to the fine topology of H. Cartan. 2000 Mathematics Subject Classification 58E20, 49N60 (primary), 58A35 (secondary).
Key Words: harmonic map • Dirichlet problem • Riemannian manifold • Riemannian polyhedron • geodesic space • Alexandrov curvature