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Carlo Mantegazza coauthored a proof of compactness for special functions of bounded variation with Giovanni Alberti. The theorem combines control of gradients with control of jump discontinuities. It allows limits of suitably bounded sequences to remain in the same function class, supporting existence arguments for variational models with interfaces.^[ref]

Giovanni Alberti coauthored a compactness theorem for special functions of bounded variation with Carlo Mantegazza. The result controls sequences that may contain jump discontinuities, using bounds on variation, gradients and jump sets. Such compactness is essential in calculus of variations problems where minimizing configurations can develop cracks or interfaces.^[ref]

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.

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,the defining condition is $D^{c}u=0$. 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.^[ref]