Bourgain--Brezis--Mironescu 公式在任意开集上的 BV 函数及其应用
Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
浏览论文内容
中文总结 AI 辅助
本文在任意开集上为 BV 函数建立 Bourgain--Brezis--Mironescu 公式，确定非局部能量 Γ-极限为全变差常数倍，并给出 BV 函数刻画及紧性结果。
中文摘要 AI 辅助
本文针对一类一般的径向磨光算子，在任意开集上建立了 BV 函数的 Bourgain--Brezis--Mironescu 公式。基于该公式，我们确定了改进的非局部能量在 L¹ 收敛意义下的 Γ-极限为全变差的常数倍。由此，我们得到了 BV 函数等价于相应非局部能量有限的刻画。我们还建立了任意开集上的局部 L¹ 紧性，以及在自然的一致可积性假设下，有界开集上的全局 L¹ 紧性。特别地，除全局紧性结果外，所有这些结果均在任意开集上成立，无需对区域的有界性、连通性或边界正则性作任何假设。反例表明，对于全局紧性，区域的有界性和一致可积性假设都是本质的。证明主要依赖于 BV 函数的一维限制以及径向磨光算子的集中论证。
英文摘要
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.
发表机构
- Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
