Measures in the dual of BV: perimeter bounds and relations with divergence-measure fields

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Giovanni E. Comi , Gian Paolo Leonardi
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引用次数: 0

Abstract

We analyze some properties of the measures in the dual of the space BV, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of BV. We exploit and refine the results of Cong Phuc and Torres (2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to BV functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for BV functions, which is of separate interest in itself and, in the context of Anzellotti’s pairing theory for divergence-measure fields, implies a new way of approximating λ-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work.
BV对偶中的度量:周界以及与发散度量场的关系
我们通过考虑满足周长约束条件的(有符号)Radon度量,分析了空间BV对偶中度量的一些性质,这意味着集合度量的绝对值受集合本身周长的控制,其总变化也属于BV的对偶。我们利用并完善了 Cong Phuc 和 Torres(2017)的成果,特别是探索了与发散度量场的关系,并证明了在给定度量的适当近似下,从集合到 BV 函数的周长约束的稳定性。作为一个重要工具,我们获得了安泽洛蒂-贾昆塔近似 BV 函数的细化,这本身就具有单独的意义,而且在安泽洛蒂的发散度量场配对理论的背景下,这意味着一种近似 λ 配对的新方法,以及它们的总变化的新边界。这些结果也适用于研究有度量数据的非参数规定均值曲率方程的弱解,这将在后续工作中探讨。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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