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Strict interior approximation of sets of finite perimeter and functions of bounded variation. (English) Zbl 1311.28005

The perimeter \(P(\Omega)\) of a measurable set \(\Omega\subset {\mathbb R}^n\) is the least upper bound for integrals \(\int_{\Omega}\operatorname{div}\phi \,dx\) taken over all \(\phi: {\mathbb R}^{n}\mapsto {\mathbb R}^{n}\) such that \(|\phi|\leq 1\) in \({\mathbb R}^{n}\). In the present paper the author investigates approximation properties of sets \(\Omega\) of finite perimeter that equals to the \((n-1)\)-dimensional Hausdorff measure of \(\partial\Omega\). He proves that an open set with this property allows strict approximation by smooth sets from within (Theorem 1.1), and any \({\mathbb R}^N\)-valued function with bounded variation on that set allows \(W^{1,1}\)-approximation.

MSC:

28A75 Length, area, volume, other geometric measure theory
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
41A63 Multidimensional problems
41A30 Approximation by other special function classes
26B15 Integration of real functions of several variables: length, area, volume
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