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A note on the theory of SBV functions. (English) Zbl 0877.49001
The paper is devoted to a new proof of the compactness theorem in SBV, originally proved by L. Ambrosio by an integralgeometric argument [Boll. Unione Mat. Ital., VII Ser. B 3, No. 4, 857-881 (1989; Zbl 0767.49001)]. As in L. Ambrosio [Calc. Var. Partial Differ. Equ. 3, No. 1, 127-137 (1995; Zbl 0837.49011)], where a similar idea was used under more restrictive assumptions, the central point of the new proof is the analysis of the distributional derivative $$D\psi(u)$$, where $$u\in \text{BV}$$ and $$\psi$$ is a Lipschitz function. Given an increasing function $$f:[0,\infty)\to [0,\infty]$$ satisfying $$f(t)/t\to\infty$$ as $$t\downarrow 0$$ and an open set $$\Omega\subset {\mathbb{R}}^n$$, it turns out that $$u\in \text{SBV}(\Omega)$$ and $$\int_{S_u}f({}u^+-u^-{}) d{\mathcal H}^{n-1}<\infty$$ if and only if $\sup_{\psi\in X(f)}D\psi(u)-\psi'(u)\lambda{}(\Omega)<\infty (1)$ for some absolutely continuous measure $$\lambda$$. Here $$X(f)$$ denotes the collection of Lipschitz and continuously differentiable functions $$\psi$$ such that ${}\psi(s)-\psi(t){}\leq f({}s-t{})\qquad\qquad\forall s, t\in {\mathbb{R}}.$ Moreover, if (1) holds, then $$\lambda$$ has the approximate differential of $$u$$ as density.
Reviewer: L.Ambrosio (Pavia)

##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 49Q20 Variational problems in a geometric measure-theoretic setting 49N60 Regularity of solutions in optimal control