Mapping problems, fundamental groups and defect measures

We study all the possible weak limits of a minimizing sequence, for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition. We show that if p is not an integer, then any such weak limit is a strong limit and, in particular, a stationary p-harmonic map which is C 1,. continuous away from a closed subset of the Hausdorff dimension . n . [p] . 1. If p is an integer, then any such weak limit is a weakly p-harmonic map along with a (n . p)-rectifiable Radon measure .. Moreover, the limiting map is C 1,. continuous away from a closed subset .=spt . . S with H n . p(S)=0. Finally, we discuss the possible varifolds type theory for Sobolev mappings.