[BibTeX]

*Published Paper*

**Inserted:** 16 may 2013

**Last Updated:** 20 feb 2015

**Journal:** Proc. Am. Math. Soc.

**Volume:** 143

**Number:** 5

**Pages:** 2069-2084

**Year:** 2015

**Links:**
Link to the published version

**Abstract:**

It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set $\Omega$ of finite perimeter in $\mathbb{R}^n$ strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the $(n{-}1)$-dimensional Hausdorff measure of the topological boundary $\partial\Omega$ equals the perimeter of $\Omega$. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of $\rm BV$-functions from a prescribed Dirichlet class.

**Tags:**
GeMeThNES

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