Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-07 , DOI: 10.1016/j.jfa.2021.109019 A.M. Caetano , D.P. Hewett , A. Moiola
We investigate two density questions for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set , is dense in whenever ∂Ω has zero Lebesgue measure and Ω is “thick” (in the sense of Triebel); and (ii) for a d-set (), is dense in whenever for some . For (ii), we provide concrete examples, for any , where density fails when and are on opposite sides of . The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether for a given closed set and . They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens. We additionally provide analogous results in the more general setting of Besov and Triebel–Lizorkin spaces.
中文翻译:
粗糙集上Sobolev,Besov和Triebel–Lizorkin空间的密度结果
我们在粗糙集上研究了Sobolev,Besov和Triebel-Lizorkin空间的两个密度问题。用最简单的Sobolev空间设置表示的主要结果是:(i)对于开放集, 密集 每当∂Ω的Lebesgue量度为零且Ω为“厚”时(按特里贝尔的意义);和(ii)为一个d -set (), 密集 每当 对于一些 。对于(ii),我们为任何,当密度失败时 和 在两边 。结果(i)和(ii)以多种方式相关,包括通过它们与是否 对于给定的封闭集 和 。它们在分形屏对声波散射的边界积分方程公式的研究中也自然而然地出现了。此外,我们在更一般的Besov和Triebel–Lizorkin空间中提供了类似的结果。