  We consider maps into Riemannian manifolds of non-positive curvature and
start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm
of such a map in terms of its energy, the $L^2$-norm of its tension field and a
topological term depending on the homotopy class. We also solve a Dirchlet
problem without an underlying variational structure, as an extension of the
topic of harmonic maps with potentials.
