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Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets
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 ΩRn, D(Ω) is dense in {uHs(Rn):suppuΩ} whenever ∂Ω has zero Lebesgue measure and Ω is “thick” (in the sense of Triebel); and (ii) for a d-set ΓRn (0<d<n), {uHs1(Rn):suppuΓ} is dense in {uHs2(Rn):suppuΓ} whenever nd2m1<s2s1<nd2m for some mN0. For (ii), we provide concrete examples, for any mN0, where density fails when s1 and s2 are on opposite sides of nd2m. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether {uHs(Rn):suppuΓ}={0} for a given closed set ΓRn and sR. 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Ω[RñdΩ 密集 {üHs[Rñ支持üΩ}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 -setΓ[Rñ0<d<ñ), {üHs1个[Rñ支持üΓ} 密集 {üHs2个[Rñ支持üΓ} 每当 -ñ-d2个--1个<s2个s1个<-ñ-d2个- 对于一些 ñ0。对于(ii),我们为任何ñ0,当密度失败时 s1个s2个 在两边 -ñ-d2个-。结果(i)和(ii)以多种方式相关,包括通过它们与是否{üHs[Rñ支持üΓ}={0} 对于给定的封闭集 Γ[Rñs[R。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。

更新日期:2021-04-14
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