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

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 Ω \Omega of finite perimeter in R n \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 ) (n{-}1) -dimensional Hausdorff measure of the topological boundary ∂ Ω \partial \Omega equals the perimeter of Ω \Omega . We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of B V BV -functions from a prescribed Dirichlet class.