Special bounded-variation space ($SBV(\Omega )$)

A BV space function is special when its derivative has no Cantor part of a bounded-variation derivative: $Du=\nabla udx+[u]{\nu }_u{\mathcal{H}}^{n-1}\lfloor J_u$. The remaining singular part is supported on the jump set of a bounded-variation function. This space accommodates sharp interfaces and is the natural setting for the Mumford–Shah functional.

For uniformly bounded values, an $L^p$ gradient bound with $p>1$, and bounded $(n-1)$-dimensional jump measure, a sequence in the SBV space admits an $L^1$-convergent subsequence whose limit remains special. Gradients converge weakly in $L^p$, and jump measure is lower semicontinuous. These hypotheses prevent diffuse singular derivatives from replacing the controlled interfaces in a minimizing limit.