Decomposition Results for Functions with Bounded Variation
Dal Maso, Gianni ; Toader, Rodica
Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 497-505 / Harvested from Biblioteca Digitale Italiana di Matematica
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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 W1;1⁢(Ω) into L1⁢(∂⁡Ω). 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 L1 and a BV function without Cantor part.

Publié le : 2008-06-01
@article{BUMI_2008_9_1_2_497_0,
     author = {Gianni Dal Maso and Rodica Toader},
     title = {Decomposition Results for Functions with Bounded Variation},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1},
     year = {2008},
     pages = {497-505},
     zbl = {1256.49056},
     mrnumber = {2424307},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_2_497_0}
}

Dal Maso, Gianni; Toader, Rodica. Decomposition Results for Functions with Bounded Variation. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 497-505. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_2_497_0/

[1] Alberti, G., A Lusin type property of gradients. J. Funct. Anal., 100 (1991), 110-118. | MR 1124295 | Zbl 0752.46025

[2] Ambrosio, L. - Coscia, A. - Dal Maso, G., Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal., 139 (1997), 201-238. | MR 1480240 | Zbl 0890.49019

[3] Ambrosio, L. - Fusco, N., - Pallara, D., Functions of bounded variation and free-discontinuity problems. Oxford Mathematical Monographs, 2000. | MR 1857292 | Zbl 0957.49001

[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. | MR 102739 | Zbl 0087.10902

[5] Giusti, E., Minimal Surfaces and Functions of Bounded Variations. Monographs in Mathematics, Birkhauser, Boston, 1984. | MR 775682 | Zbl 0545.49018

[6] Temam, R., Problèmes mathématiques en plasticité. Gauthier-Villars, Paris, 1983. | MR 711964 | Zbl 0547.73026