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.

A standard sufficient form uses a bounded Lipschitz domain $\Omega$, scalar functions $u_k\in SBV(\Omega )$, a uniform essential supremum bound, and some $p>1$ such thatThere is a subsequence converging in L1 norm to a function $u\in SBV(\Omega )$. Its approximate gradients converge weakly in $L^p$, and the jump-set measure is bounded by the lower limit of the approximants' jump-set measures. The theorem keeps the Cantor part of a BV derivative out of the limit. ^[ref]

The hypotheses do distinct jobs. The value bound and jump-size bound control jump variation; $p>1$ gives uniform integrability of the absolutely continuous derivatives. Ordinary bounded-variation compactness alone permits a diffuse singular part in the limit. Domain or tightness control also matters for global convergence: translating a fixed interval indicator arbitrarily far along $\mathbb{R}$ keeps all these local size bounds but gives no strongly convergent global $L^1$ subsequence.