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arXiv · 2609.27163

Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications

Jiang Li·Zhuang Wang
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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 \(L^1\)-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.

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Jiang Li, Zhuang Wang. 2026-09-22. Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications. https://arxiv.org/abs/2609.27163

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