Calculus of Variations and Geometric Measure Theory

T. Schmidt

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

created by schmidt on 16 May 2013
modified on 20 Feb 2015


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


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