Another Look at Elliptic Homogenization

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-12-03 DOI:10.1007/s00032-023-00389-y

Andrea Braides, Giuseppe Cosma Brusca, Davide Donati

引用次数: 1

Abstract

We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as \(s\rightarrow 1\). In a seminal paper by Bourgain et al. (Another look at Sobolev spaces. In: Optimal control and partial differential equations. IOS, Amsterdam, pp 439–455, 2001) it is proven that if the coefficient is constant then this sequence \(\Gamma \)-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by \(\varepsilon \) the scale of the oscillations and we assume that \(1-s<\!<\varepsilon ^2\), this sequence converges to the homogenized functional formally obtained by separating the effects of s and \(\varepsilon \); that is, by the homogenization as \(\varepsilon \rightarrow 0\) of the Dirichlet integral with oscillating coefficient obtained by formally letting \(s\rightarrow 1\) first.

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对椭圆均匀化的另一个看法

我们考虑具有振荡系数的归一化(s, 2)-Gagliardo半模序列的极限为\(s\rightarrow 1\)。在布尔甘等人的一篇开创性论文中(另一种对索博列夫空间的看法)。内:最优控制和偏微分方程。IOS, Amsterdam, pp 439-455, 2001)证明了如果系数是常数，那么这个数列\(\Gamma \) -收敛于Dirichlet积分的一个倍数。这里我们证明了，如果我们用\(\varepsilon \)表示振荡的尺度并且我们假设\(1-s<\!<\varepsilon ^2\)，这个序列收敛于通过分离s和\(\varepsilon \)的影响而得到的形式的均匀泛函;即将形式上先让\(s\rightarrow 1\)得到的带振荡系数的狄利克雷积分均化为\(\varepsilon \rightarrow 0\)。

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.