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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 $D\psi \left(u\right)$, where $u\in \text{BV}$ and $\psi$ is a Lipschitz function. Given an increasing function $f:\left[0,\infty \right)\to \left[0,\infty \right]$ satisfying $f\left(t\right)/t\to \infty$ as $t↓0$ and an open set ${\Omega }\subset {ℝ}^{n}$, it turns out that $u\in \text{SBV}\left({\Omega }\right)$ and ${\int }_{{S}_{u}}f\left({u}^{+}-{u}^{-}\right)d{ℋ}^{n-1}<\infty$ if and only if

$\underset{\psi \in X\left(f\right)}{sup}D\psi \left(u\right)-{\psi }^{\text{'}}\left(u\right)\lambda \left({\Omega }\right)<\infty \phantom{\rule{2.em}{0ex}}\left(1\right)$

for some absolutely continuous measure $\lambda$. Here $X\left(f\right)$ denotes the collection of Lipschitz and continuously differentiable functions $\psi$ such that

$\psi \left(s\right)-\psi \left(t\right)\le f\left(s-t\right)\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\forall s,t\in ℝ·$

Moreover, if (1) holds, then $\lambda$ has the approximate differential of $u$ as density.

MSC:
 49J10 Free problems in several independent variables (existence) 49Q20 Variational problems in a geometric measure-theoretic setting 49N60 Regularity of solutions in calculus of variations