Calculus of Variations and Geometric Measure Theory
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G. Dal Maso - R. Toader

Decomposition results for functions with bounded variation

created by dalmaso on 23 Sep 2008

[BibTeX]

Published Paper

Inserted: 23 sep 2008

Journal: Boll. Un. Mat. Ital. (9)
Volume: 1
Pages: 497-505
Year: 2008

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 ingular $BV$ 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.


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