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Extensions and traces of BV functions in rough domains and generalized Cheeger sets

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Abstract

Via the method of interior approximation, we prove trace and extension results for BV functions defined on a bounded domain \(\Omega \subset {\mathbb {R}}^n\) such that \(\Omega \) satisfies

$$\begin{aligned} {\mathscr {H}}^{n-1}(\partial \Omega \setminus \Omega ^0)<\infty , \end{aligned}$$
(0.1)

where \({\mathscr {H}}^{n-1}\) is the \((n-1)\) dimensional Hausdorff measure and \(\Omega ^0\) is the measure-theoretic exterior of \(\Omega \). Stronger results are obtained on a subclass of such domains, which are outward minimizing domains. We also obtain new weak regularity results for generalized Cheeger sets as byproducts.

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Acknowledgements

C. Gui is partially supported by NSF grants DMS-1901914 and DMS-2155183. Y. Hu is partially supported by NSFC NO.12101612 and NSFC NO.12171456. Research of Qinfeng Li is supported by the National Science Fund for Youth Scholars (No. 1210010723) and the Fundamental Research Funds for the Central Universities, Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics.

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Authors and Affiliations

  1. The University of Texas at San Antonio, San Antonio, TX, USA

    Changfeng Gui

  2. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, 410083, P. R. China

    Changfeng Gui & Yeyao Hu

  3. Hunan University, Changsha, Hunan, China

    Qinfeng Li

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  1. Changfeng Gui
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  2. Yeyao Hu
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Correspondence to Changfeng Gui.

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Communicated by L. Szekelyhidi.

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Gui, C., Hu, Y. & Li, Q. Extensions and traces of BV functions in rough domains and generalized Cheeger sets. Calc. Var. 62, 38 (2023). https://doi.org/10.1007/s00526-022-02377-3

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