Density results for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets

Abstract

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 $Ω\subset\mathbb R^n$, $\mathcal{D}(Ω)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overlineΩ\}$ whenever $\partialΩ$ has zero Lebesgue measure and $Ω$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $Γ\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset Γ\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset Γ\}$ whenever $-\frac{n-d}{2}-m-1<s_{2}\leq s_{1}<-\frac{n-d}{2}-m$ for some $m\in\mathbb N_0$. For (ii), we provide concrete examples, for any $m\in\mathbb N_0$, where density fails when $s_1$ and $s_2$ are on opposite sides of $-\frac{n-d}{2}-m$. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset Γ\}=\{0\}$ for a given closed set $Γ\subset\mathbb R^n$ and $s\in \mathbb R$. 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.