On smooth interior approximation of sets of finite perimeter
HTML articles powered by AMS MathViewer
- 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
- HTML | PDF | Request permission
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
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Elisabetta Barozzi, Eduardo Gonzalez, and Umberto Massari, Pseudoconvex sets, Ann. Univ. Ferrara Sez. VII Sci. Mat. 55 (2009), no. 1, 23–35. MR 2506062, DOI 10.1007/s11565-009-0063-7
- Gui-Qiang G. Chen, Giovanni E. Comi, and Monica Torres, Cauchy fluxes and Gauss-Green formulas for divergence-measure fields over general open sets, Arch. Ration. Mech. Anal. 233 (2019), no. 1, 87–166. MR 3974639, DOI 10.1007/s00205-018-01355-4
- Gui-Qiang Chen and Hermano Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89–118. MR 1702637, DOI 10.1007/s002050050146
- Gui-Qiang Chen, Qinfeng Li, and Monica Torres, Traces and extensions of bounded divergence-measure fields on rough open sets, Indiana Univ. Math. J. 69 (2020), no. 1, 229–264. MR 4077162, DOI 10.1512/iumj.2020.69.8375
- Gui-Qiang Chen and Monica Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 245–267. MR 2118477, DOI 10.1007/s00205-004-0346-1
- Gui-Qiang Chen, Monica Torres, and William P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. 62 (2009), no. 2, 242–304. MR 2468610, DOI 10.1002/cpa.20262
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Giovanni E. Comi and Monica Torres, One-sided approximation of sets of finite perimeter, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 1, 181–190. MR 3621776, DOI 10.4171/RLM/757
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Changfeng Gui, Yeyao Hu, and Qinfeng Li, Extensions and traces of BV functions in rough domains and generalized Cheeger sets, Calc. Var. Partial Differential Equations 62 (2023), no. 2, Paper No. 38, 19. MR 4525719, DOI 10.1007/s00526-022-02377-3
- Gian Paolo Leonardi and Giorgio Saracco, The prescribed mean curvature equation in weakly regular domains, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Paper No. 9, 29. MR 3767675, DOI 10.1007/s00030-018-0500-3
- Qinfeng Li and Monica Torres, Morrey spaces and generalized Cheeger sets, Adv. Calc. Var. 12 (2019), no. 2, 111–133. MR 3935853, DOI 10.1515/acv-2016-0050
- Francesco Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. MR 2976521, DOI 10.1017/CBO9781139108133
- Thomas Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2069–2084. MR 3314116, DOI 10.1090/S0002-9939-2014-12381-1
- Christoph Scheven and Thomas Schmidt, BV supersolutions to equations of 1-Laplace and minimal surface type, J. Differential Equations 261 (2016), no. 3, 1904–1932. MR 3501836, DOI 10.1016/j.jde.2016.04.015
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