  In this paper we prove the existence of a solution to the Dirichlet problem
for harmonic maps into a geodesic ball on which the squared distance function
from the origin is strictly convex. This improves a celebrated theorem obtained
by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature
assumptions on the target are required. Our proof relies on a careful analysis
of the Sobolev spaces of maps involved in the variational process, and on a
deformation result which permits to glue a suitable Euclidean end to the
geodesic ball.
