On the Dirichlet problem for p-harmonic maps I: compact targets

In this paper we solve the relative homotopy Dirichlet problem for p-harmonic maps from compact manifolds with boundary to compact manifolds of non-positive sectional curvature. The proof, which is based on the direct calculus of variations, uses some ideas of B. White to define the relative d-homotopy type of Sobolev maps. One of the main points of the proof consists in showing that the regularity theory by Hardt and Lin can be applied. A comprehensive uniqueness result for general complete targets with non-positive curvature is also given.