Weakly Differentiable Functions on Varifolds
- 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 Poincare-type embeddings, embeddings into spaces of continuous and sometimes Holder-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 varifoldsThe 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 Poincare-type embeddings, embeddings into spaces of continuous and sometimes Holder-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.…
Verfasserangaben: | Ulrich MenneORCiDGND |
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DOI: | https://doi.org/10.1512/iumj.2016.65.5829 |
ISSN: | 0022-2518 |
ISSN: | 1943-5258 |
Titel des übergeordneten Werks (Englisch): | Indiana University mathematics journal |
Verlag: | Indiana University, Department of Mathematics |
Verlagsort: | Bloomington |
Publikationstyp: | Wissenschaftlicher Artikel |
Sprache: | Englisch |
Jahr der Erstveröffentlichung: | 2016 |
Erscheinungsjahr: | 2016 |
Datum der Freischaltung: | 22.03.2020 |
Freies Schlagwort / Tag: | (generalised) wealdy differentiable function; Rectifiable varifold; Sobolev Poincare inequality; approximate differentiability; coarea formula; curvature varifold; decomposition; distributional boundary; geodesic distance; relative isoperimetric inequality |
Band: | 65 |
Seitenanzahl: | 112 |
Erste Seite: | 977 |
Letzte Seite: | 1088 |
Organisationseinheiten: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
Peer Review: | Referiert |