Weakly differentiable functions on varifolds

Ulrich Menne*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities for certain compositions with smooth functions. In this class, the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincaré-type embeddings, embeddings into spaces of continuous and sometimes Hölder-continuous functions, and point-wise differentiability results both of approximate and integral type as well as coarea formulae. As a prerequisite for this study, decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained. Indiana University Mathematics Journal

Original languageEnglish
Pages (from-to)977-988
Number of pages12
JournalIndiana University Mathematics Journal
Volume65
Issue number3
DOIs
Publication statusPublished - 2016
Externally publishedYes

Keywords

  • (generalised) weakly differentiable function
  • Approximate differentiability
  • Coarea formula
  • Curvature varifold
  • Decomposition
  • Distributional boundary
  • Geodesic distance
  • Rectifiable varifold
  • Relative isoperimetric inequality
  • Sobolev Poincaréinequality

ASJC Scopus subject areas

  • Mathematics(all)

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