Strict interior approximation of sets of finite perimeter and functions of bounded variation

Schmidt Thomas

year: 2015
journal: Proc. Am. Math. Soc.
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.

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