SBV compactness theorem

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.