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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On smooth interior approximation of sets of finite perimeter
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by Changfeng Gui, Yeyao Hu and Qinfeng Li
Proc. Amer. Math. Soc. 151 (2023), 1949-1962
DOI: https://doi.org/10.1090/proc/15640
Published electronically: February 17, 2023

Abstract:

In this paper, we prove that for any bounded set of finite perimeter $\Omega \subset \mathbb {R}^n$, we can choose smooth sets $E_k \Subset \Omega$ such that $E_k \rightarrow \Omega$ in $L^1$ and \begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega )+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In the above $\Omega ^1$ is the measure-theoretic interior of $\Omega$, $P(\cdot )$ denotes the perimeter functional on sets, and $C_1(n)$ is a dimensional constant.

Conversely, we prove that for any sets $E_k \Subset \Omega$ satisfying $E_k \rightarrow \Omega$ in $L^1$, there exists a dimensional constant $C_2(n)$ such that the following inequality holds: \begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega )+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In particular, these results imply that for a bounded set $\Omega$ of finite perimeter, \begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*} holds if and only if there exists a sequence of smooth sets $E_k$ such that $E_k \Subset \Omega$, $E_k \rightarrow \Omega$ in $L^1$ and $P(E_k) \rightarrow P(\Omega )$.

References
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Bibliographic Information
  • Changfeng Gui
  • Affiliation: Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau, China; and Department of Mathematics, University of Texas at San Antonio, Texas 78255
  • MR Author ID: 326332
  • ORCID: 0000-0001-5903-6188
  • Email: changfeng.gui@utsa.edu
  • Yeyao Hu
  • Affiliation: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, People’s Republic of China
  • MR Author ID: 1202002
  • Email: huyeyao@gmail.com
  • Qinfeng Li
  • Affiliation: School of Mathematics, Hunan University, Changsha, Hunan, People’s Republic of China
  • MR Author ID: 1036573
  • Email: liqinfeng1989@gmail.com
  • Received by editor(s): September 20, 2020
  • Received by editor(s) in revised form: February 22, 2021
  • Published electronically: February 17, 2023
  • Additional Notes: The research of the first author was partially supported by NSF grants DMS-1901914 and DMS-2155183; The research of the second author was partially supported by NSFC grants No. 12101612 and No. 12171456; The research of the third author was supported by National Science Fund for Youth Scholars (No. 12101215).
    The third author is the corresponding author.
  • Communicated by: Ryan Hynd
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 1949-1962
  • MSC (2020): Primary 28A75, 46E35, 49Q15, 49Q20
  • DOI: https://doi.org/10.1090/proc/15640
  • MathSciNet review: 4556191