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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)

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