Maps with prescribed tension fields

Abstract

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.