TY - JOUR

T1 - Mapping Problems, Fundamental Groups and Defect Measures

AU - Lin, Fanghua

N1 - Funding Information:
Received September 5, 1998, Accepted September 20, 1998 Partially supported by NSF Grant DMS 9626166
Funding Information:
The research is partially supported by an NSF Grant DMS # 9706862. Part of the work was completed while the author was visiting the Max-Planck Institute of Mathematics in Sciences at Leipzig. The author wishes to thank Professor J. Jost for the invitation and the warm hospitality.

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

KW - Defect measure

KW - Generalized varifold

KW - Harmonic mapping

KW - Rectifiability

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U2 - 10.1007/s10114-999-0059-3

DO - 10.1007/s10114-999-0059-3

M3 - Article

AN - SCOPUS:0002557930

VL - 15

SP - 25

EP - 52

JO - Acta Mathematica Sinica, English Series

JF - Acta Mathematica Sinica, English Series

SN - 1439-8516

IS - 1

ER -