TY  - JOUR
A1  - Menne, Ulrich
T1  - Weakly Differentiable Functions on Varifolds
JF  - Indiana University mathematics journal
N2  - 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 varifolds are obtained.
KW  - Rectifiable varifold
KW  - (generalised) wealdy differentiable function
KW  - distributional boundary
KW  - decomposition
KW  - relative isoperimetric inequality
KW  - Sobolev Poincare inequality
KW  - approximate differentiability
KW  - coarea formula
KW  - geodesic distance
KW  - curvature varifold
Y1  - 2016
U6  - https://doi.org/10.1512/iumj.2016.65.5829
SN  - 0022-2518
SN  - 1943-5258
VL  - 65
SP  - 977
EP  - 1088
PB  - Indiana University, Department of Mathematics
CY  - Bloomington
ER  -