*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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