Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global $s$-minimal surfaces

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2016-12-25 DOI:10.4171/IFB/402

L. Lombardini

引用次数: 20

Abstract

In the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global $s$-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e. local and global), showing that in bounded open sets with Lipschitz boundary they coincide. However, in general this is not true in unbounded open sets, where a global $s$-minimal set may fail to exist (we provide an example in the case of a cylinder $\Omega\times\mathbb{R}$).

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用光滑集逼近有限分数周长集及局部与全局最小曲面的比较

在本文的第一部分中，我们证明了一个集$E$具有局部有限的$s$周长当且仅当它能被光滑开集在适当意义上近似。第二部分证明了局部和全局$s$-极小集的存在性和紧性等基本性质。我们还比较了两种最小器的概念(即局部和全局)，表明在具有Lipschitz边界的有界开集中它们是重合的。然而，一般来说，这在无界开集中是不成立的，在无界开集中，全局$s$-最小集可能不存在(我们提供了一个圆柱体$\Omega\乘以\mathbb{R}$的例子)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

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.