Strict interior approximation of sets of finite perimeter and functions of bounded variation
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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.References
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Additional Information
- Thomas Schmidt
- Affiliation: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy – and – Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
- Email: thomas.schmidt@sns.it, thomas.schmidt@math.uzh.ch
- Received by editor(s): May 24, 2013
- Received by editor(s) in revised form: October 9, 2013
- Published electronically: November 25, 2014
- Additional Notes: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement GeMeThnES No. 246923.
- Communicated by: Tatiana Toro
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 2069-2084
- MSC (2010): Primary 28A75, 26B30, 41A63, 41A30; Secondary 28A78, 26B15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12381-1
- MathSciNet review: 3314116