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

Schmidt, Thomas

doi:10.1090/S0002-9939-2014-12381-1

Strict interior approximation of sets of finite perimeter and functions of bounded variation
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Proc. Amer. Math. Soc. 143 (2015), 2069-2084 Request permission

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 $BV$-functions from a prescribed Dirichlet class.

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