Density results for Sobolev, Besov and Triebel-Lizorkin spaces on rough sets
Density results for Sobolev, Besov and Triebel-Lizorkin spaces on rough sets
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粗糙集上 Sobolev、Besov 和 Triebel-Lizorkin 空间的密度结果
DOI：
10.1016/j.jfa.2021.109019
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发表时间：
2021
影响因子：
1.7
通讯作者：
Caetano A
中科院分区：
文献类型：
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作者：
Caetano A
研究了粗糙集上Sobolev、Besov和Triebel-Lizorkin空间的两个密度问题。在最简单的Sobolev空间中，我们的主要结果是：（i）对于开集Ω <$Rn，当Ω有零Lebesgue测度且Ω是“厚的”时，D（Ω）在{u∈ Hs（Rn）：suppu <$Ω <$}中是稠密的（在Triebel意义下）;（ii）对d-集Γ <$Rn（0< d< n），{u∈ Hs 1（Rn）：suppu <$Γ}在{u∈ Hs 2（Rn）：suppu <$r}当− n− d 2− m− 1< s 2≤ s 1<− n− d 2− m，其中m∈ N 0。对于（ii），我们提供了具体的例子，对于任何m∈ N 0，当s1和s2在− n-d 2− m的相对两侧时，密度失效。结果（i）和（ii）在许多方面都是相关的，包括通过它们与对于给定的闭集r R n和s∈ R是否{u∈ H s（R n）：supp u r}={0}的问题的联系。它们也都自然地出现在分形屏幕的声波散射的边界积分方程公式的研究。我们还提供了类似的结果，在更一般的设置Besov和Triebel-Lizorkin空间。
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 Ω⊂ R n, D (Ω) is dense in {u∈ H s (R n): supp u⊂ Ω‾} whenever∂ Ω has zero Lebesgue measure and Ω is “thick”(in the sense of Triebel); and (ii) for a d-set Γ⊂ R n (0< d< n),{u∈ H s 1 (R n): supp u⊂ Γ} is dense in {u∈ H s 2 (R n): supp u⊂ Γ} whenever− n− d 2− m− 1< s 2≤ s 1<− n− d 2− m for some m∈ N 0. For (ii), we provide concrete examples, for any m∈ N 0, where density fails when s 1 and s 2 are on opposite sides of− n− d 2− m. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether {u∈ H s (R n): supp u⊂ Γ}={0} for a given closed set Γ⊂ R n and s∈ R. 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.