Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
Abstract
In this paper, we establish a Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the Γ-limit of the improved nonlocal energies with respect to L1-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local L1-compactness on arbitrary open sets and global L1-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.
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BibTeX
@misc{li2026bourgainbrezismironescu,
title = {{Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications}},
author = {Jiang Li and Zhuang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27163},
eprint = {2609.27163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27163},
}
RIS
TY - UNPB
TI - Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
AU - Jiang Li
AU - Zhuang Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27163
UR - https://arxiv.org/abs/2609.27163
AB - In this paper, we establish a Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets for a general class of radial mollifiers. Building on this formula, we identify the Γ-limit of the improved nonlocal energies with respect to L^1-convergence as a constant multiple of the total variation. As a consequence, we obtain a characterization of BV functions in terms of the finiteness of the corresponding nonlocal energies. We also establish local L^1-compactness on arbitrary open sets and global L^1-compactness on bounded open sets under a natural uniform integrability assumption. In particular, except for the global compactness result, all these results hold on arbitrary open sets, with no assumptions on boundedness, connectedness, or boundary regularity. Counterexamples show that for global compactness, both the boundedness of the domain and the uniform integrability assumption are essential. The proofs rely primarily on one-dimensional restrictions of BV functions and a concentration argument for radial mollifiers.
ER -