SBV functions over a rectifiable current and a compactness theorem
type: Item
instance of: scholarly article
title: SBV functions over a rectifiable current and a compactness theorem
zbMATH Open document ID: 0877.49002
DOI: 10.1016/S0021-7824(97)89943-6
author: Elisabetta Ossanna
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1997-11-09T00:00:00Z
review text: In this paper special functions with bounded variation are introduced in a very general geometric setting, assuming that the domain of the functions is the support \(M\subset{\mathbb{R}}^n\) of an integral current. The paper is the natural continuation of \textit{G. Anzellotti, S. Delladio} and \textit{G. Scianna} [Ann. Mat. Pura Appl., IV. Ser. 170, 257-296 (1996)], where BV functions on \(M\) are defined as those functions whose subgraph \[ E_u:=\{(x,t)\in M\times{\mathbb{R}}:0<t<u(x)\text{ or }u(x)<t<0\} \] (with a suitable orientation induced by the one of \(M\)) has boundary, in the sense of currents, with finite mass. By Federer--Fleming theorem, it is possible in this way to show that BV functions have a \({\mathcal H}^n\)-rectifiable graph \(\Gamma_u\). By looking at the projection \(\pi(\Gamma^*_u)\) on \(M\) of the vertical part \(\Gamma^*_u\) of \(\Gamma_u\), SBV functions can be defined as those BV functions such that \(\pi(\Gamma^*_u)\) is countably \({\mathcal H}^{n-1}\)-rectifiable (this set can also be identified, up to \({\mathcal H}^{n-1}\)-negligible sets, with the approximate discontinuity set of \(u\)). Following basically the strategy of \textit{L. Ambrosio} [Calc. Var. Partial Differ. Equ. 3, No. 1, 127-137 (1995; Zbl 0837.49011)] in this more general setting, a generalized compactness theorem in SBV is proved.
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49N60
zbMATH DE Number: 1010945
zbMATH Keywords: special functions of bounded variation,
zbMATH Keywords: currents
zbMATH Keywords: geometric measure theory
zbMATH Keywords: SBV
zbMATH Keywords: compactness theorem
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
cites work: Q4710887
cites work: Existence theory for a new class of variational problems
cites work: Variational problems in SBV and image segmentation
cites work: A new proof of the SBV compactness theorem
cites work: Q3970542
cites work: BV functions over rectifiable currents
cites work: Q5749875
cites work: Functionals defined on measures and applications to non equi-uniformly elliptic problems
cites work: Q3202752
cites work: Existence theorem for a minimum problem with free discontinuity set
cites work: Oriented and nonoriented curvature varifolds
cites work: Q5565773
cites work: Q5186292
cites work: Variational problems with free interfaces
cites work: Optimal approximations by piecewise smooth functions and associated variational problems
cites work: Q3336268
cites work: Q3862204
cites work: Drops of nematic liquid crystals
Recommended article: Q4332717
Recommended article: Q4954179
Recommended article: Q3972957
Recommended article: Q3732444
Recommended article: Lifting of BV functions with values in \(S^1\)
Recommended article: Rectifiability of the distributional Jacobian for a class of functions
Recommended article: Q3211883
Recommended article: Weighted BV functions
Recommended article: On SBV dual
Recommended article: \(\Gamma\)-convergence, Sobolev norms, and BV functions
description: scientific article; zbMATH DE number 1010945