# Decomposition Results for Functions with Bounded Variation

• Volume: 1, Issue: 2, page 497-505
• ISSN: 0392-4041

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## Abstract

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Some decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from $W^{1;1}(\Omega)$ into $L^{1}(\partial\Omega)$. More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely singular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue measure). It can also be decomposed as the sum of a purely singular BV function and a BV function without Cantor part. We also prove similar results for the space BD of functions with bounded deformation. In particular, we show that every BD function can be written as the sum of a BD function without jumps and a BV function without Cantor part. Therefore, every BD function without Cantor part is the sum of a function whose symmetrized gradient belongs to $L^{1}$ and a BV function without Cantor part.

## How to cite

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Dal Maso, Gianni, and Toader, Rodica. "Decomposition Results for Functions with Bounded Variation." Bollettino dell'Unione Matematica Italiana 1.2 (2008): 497-505. <http://eudml.org/doc/290462>.

@article{DalMaso2008,
abstract = {Some decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from $W^\{1;1\}(\Omega)$ into $L^1(\partial \Omega)$. More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely singular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue measure). It can also be decomposed as the sum of a purely singular BV function and a BV function without Cantor part. We also prove similar results for the space BD of functions with bounded deformation. In particular, we show that every BD function can be written as the sum of a BD function without jumps and a BV function without Cantor part. Therefore, every BD function without Cantor part is the sum of a function whose symmetrized gradient belongs to $L^1$ and a BV function without Cantor part.},
author = {Dal Maso, Gianni, Toader, Rodica},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {497-505},
publisher = {Unione Matematica Italiana},
title = {Decomposition Results for Functions with Bounded Variation},
url = {http://eudml.org/doc/290462},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Dal Maso, Gianni
TI - Decomposition Results for Functions with Bounded Variation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/6//
PB - Unione Matematica Italiana
VL - 1
IS - 2
SP - 497
EP - 505
AB - Some decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from $W^{1;1}(\Omega)$ into $L^1(\partial \Omega)$. More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely singular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue measure). It can also be decomposed as the sum of a purely singular BV function and a BV function without Cantor part. We also prove similar results for the space BD of functions with bounded deformation. In particular, we show that every BD function can be written as the sum of a BD function without jumps and a BV function without Cantor part. Therefore, every BD function without Cantor part is the sum of a function whose symmetrized gradient belongs to $L^1$ and a BV function without Cantor part.
LA - eng
UR - http://eudml.org/doc/290462
ER -

## References

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1. ALBERTI, G., A Lusin type property of gradients. J. Funct. Anal., 100 (1991), 110-118. Zbl0752.46025MR1124295DOI10.1016/0022-1236(91)90104-D
2. AMBROSIO, L. - COSCIA, A. - DAL MASO, G., Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal., 139 (1997), 201-238. Zbl0890.49019MR1480240DOI10.1007/s002050050051
3. AMBROSIO, L. - FUSCO, N., - PALLARA, D., Functions of bounded variation and free-discontinuity problems. Oxford Mathematical Monographs, 2000. Zbl0957.49001MR1857292
4. GAGLIARDO, E., Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova, 27 (1957) 284-305. Zbl0087.10902MR102739
5. GIUSTI, E., Minimal Surfaces and Functions of Bounded Variations. Monographs in Mathematics, Birkhauser, Boston, 1984. Zbl0545.49018MR775682DOI10.1007/978-1-4684-9486-0
6. TEMAM, R., Problèmes mathématiques en plasticité. Gauthier-Villars, Paris, 1983. Zbl0547.73026MR711964

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