  In this two papers we deal with the relative homotopy Dirichlet problem for
p-harmonic maps from compact manifolds with boundary to manifolds of
non-positive sectional curvature. Notably, we give a complete solution to the
problem in case the target manifold is either compact and a new proof in case
it is rotationally symmetric or two dimensional and simply connected. The proof
of the compact case uses some ideas of White to define the relative d-homotopy
type of Sobolev maps, and the regularity theory by Hardt and Lin. To deal with
non-compact targets we introduce a periodization procedure which permits to
reduce the problem to the previous one. Also, a general uniqueness result is
given.
