Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

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


Author: Thomas Schmidt
Journal: Proc. Amer. Math. Soc. 143 (2015), 2069-2084
MSC (2010): Primary 28A75, 26B30, 41A63, 41A30; Secondary 28A78, 26B15
Published electronically: November 25, 2014
MathSciNet review: 3314116
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28A75, 26B30, 41A63, 41A30, 28A78, 26B15

Retrieve articles in all journals with MSC (2010): 28A75, 26B30, 41A63, 41A30, 28A78, 26B15


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

DOI: https://doi.org/10.1090/S0002-9939-2014-12381-1
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.