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MiS Preprint
47/1998
Mapping problems, fundamental groups and defect measures
Fanghua Lin
Abstract
We study all 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 $\mathbb{C}^{1,a}$ continuous away from a closed subset of the Hausdorff dimension $\leq n - [p] -1$. If p is an integer, then any such weak limit is a weakly p-harmonic map along with a (n-p)-rectifiable Radon measure $\mu$. Moreover, the limiting map is $\mathbb{C}^{1,a}$ continuous away from a closed subset $\Sigma =spt \mu \cup S$ with $H^{n-p} (S) = 0$. Finally, we discussed the possible varifolds type theory for Sobolev mappings.