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,α 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 μ. Moreover, the limiting map is C^1,α continuous away from a closed subset Σ = spt μ ∪ S with H^n-p(S) = 0. Finally, we discuss the possible varifolds type theory for Sobolev mappings.