= Special function of bounded variation
{title2=$SBV(\Omega)$}
= Special functions of bounded variation
{synonym}
A <special function of bounded variation> is a <function of bounded variation on a domain> whose <distributional derivative> has no diffuse singular part. Writing the derivative measure as an absolutely continuous part, a jump part and a remaining Cantor part,
$$
Du=\nabla u\,\mathcal L^n+(u^+-u^-)\nu_u\,\mathcal H^{n-1}\!\restriction J_u+D^cu,
$$
the defining condition is $D^cu=0$. The space is denoted $SBV(\Omega)$. Here $J_u$ is the jump set and $u^\pm$ are the two traces across it; $\mathcal L^n$ and $\mathcal H^{n-1}$ are <Lebesgue measure> and <Hausdorff measure>.
On an interval containing zero, $u=1_{(0,\infty)}$ has derivative a <Dirac measure> at zero and belongs to $SBV$: jumps are allowed. The <Cantor function> instead has a diffuse singular derivative and is not in $SBV$. Thus “special” separates sharp interfaces from more distributed singular variation. This makes $SBV$ useful for <image segmentation> and variational models with both bulk and interface energies.<A Note on the Theory of SBV Functions by Giovanni Alberti and Carlo Mantegazza>[Introduction and Definition 1.3]{ref}
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