Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-08-20 DOI:10.1016/j.na.2024.113642
Carolin Kreisbeck, Hidde Schönberger
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引用次数: 0

Abstract

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via Γ-convergence as the fractional parameter tends to 1.

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非局部梯度为零的非常数函数及其在非局部新曼类问题中的作用
这项研究围绕非局部梯度(更准确地说,是有限域分数梯度)消失的函数的性质和应用展开。令人惊奇的是,与经典的局部理论不同,我们发现这类函数构成了一个无限维的向量空间。我们的主要结果通过两个简单的特征来描述非局部梯度为零的函数,即它们在边界周围层中的值及其平均值。证明利用了伪微分算子边界值问题求解理论的最新进展。我们对这些发现进行了补充,讨论了此类函数的正则特性,并给出了示例。关于应用,我们提供了几种有用的技术工具,用于在放弃常见补值条件的情况下处理非局部索波列夫空间。其中包括新的非局部 Poincaré 不等式和紧凑性声明,这些都是在剔除非局部梯度消失的函数后得到的。根据变分法,我们利用之前的发现研究了一类受自然边界条件限制的非局部偏微分方程,特别是非局部 Neumann 型问题。我们的分析包括对良好求解性的证明,以及当分数参数趋向于 1 时,通过Γ收敛与经典局部对应方程的严格联系。
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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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