分数索波列夫空间和分数变化的分布方法:渐近学 I.

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Accounts of Chemical Research Pub Date : 2023-01-01 Epub Date: 2022-06-20 DOI:10.1007/s13163-022-00429-y
Giovanni E Comi, Giorgio Stefani
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引用次数: 0

摘要

我们继续研究我们之前的工作 (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019) 中引入的 R n 中阶 α∈ ( 0 , 1 ) 的有界分数变化函数空间 B V α ( R n ) 。在对 Comi 和 Stefani (2019) 的某些结果做了一些技术上的改进之后(这些结果可能会引起一些单独的兴趣),我们讨论了所涉及的分数算子在 α → 1 - 时的渐近行为。我们证明,当 α → 1 - 时,W 1 , p 函数的 α 梯度在 L p 中收敛于所有 p∈ [ 1 , + ∞ ) 的梯度。此外,我们证明当 α → 1 - 时,分数 α 变量在点上和Γ - 极限意义上都收敛于标准的 De Giorgi 变量。最后,我们证明,对于任何给定的 α∈ ( 0 , 1 ) ,分式 β 变量在 β → α - 时都会在点上和 Γ - 极限意义上收敛于分式 α 变量。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I.

We continue the study of the space  B V α ( R n ) of functions with bounded fractional variation in  R n of order α ( 0 , 1 ) introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as α 1 - . We prove that the α -gradient of a W 1 , p -function converges in L p to the gradient for all p [ 1 , + ) as α 1 - . Moreover, we prove that the fractional α -variation converges to the standard De Giorgi's variation both pointwise and in the Γ -limit sense as α 1 - . Finally, we prove that the fractional β -variation converges to the fractional α -variation both pointwise and in the Γ -limit sense as β α - for any given α ( 0 , 1 ) .

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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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