Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows

Fang-Hua Lin, ChangYou Wang

Research output: Contribution to journalArticle

Abstract

We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke's flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold.

Original languageEnglish (US)
Pages (from-to)209-259
Number of pages51
JournalAnnales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
Volume19
Issue number2
DOIs
StatePublished - 2002

Fingerprint

Harmonic Maps
Defects
Harmonic
Limiting
Suitable Weak Solutions
Convergent Sequence
Heat Flow
Stratification
Quantization
Space-time
Verify
Heat transfer
Energy

Keywords

  • Brakke's flow
  • Concentration measures
  • Energy quantization
  • Generalized varifold flows
  • Harmonic or approximated harmonic map flows
  • Rectifiablity
  • Stratification

ASJC Scopus subject areas

  • Analysis

Cite this

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abstract = "We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke's flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold.",
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AU - Lin, Fang-Hua

AU - Wang, ChangYou

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AB - We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke's flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold.

KW - Brakke's flow

KW - Concentration measures

KW - Energy quantization

KW - Generalized varifold flows

KW - Harmonic or approximated harmonic map flows

KW - Rectifiablity

KW - Stratification

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