Mapping Problems, Fundamental Groups and Defect Measures

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Abstract

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 C1,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 C1,α continuous away from a closed subset Σ = spt μ ∪ S with Hn-p(S) = 0. Finally, we discuss the possible varifolds type theory for Sobolev mappings.

Original languageEnglish (US)
Pages (from-to)25-52
Number of pages28
JournalActa Mathematica Sinica, English Series
Volume15
Issue number1
DOIs
StatePublished - Jan 1 1999

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Keywords

  • Defect measure
  • Generalized varifold
  • Harmonic mapping
  • Rectifiability

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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