Properties of IFS attractors with non-empty interiors, related rough domains, and associated function spaces and scattering problems
Authors
António Caetano
Simon N. Chandler-Wilde
David P. Hewett
Abstract
We study fractal sets $Γ\subset \mathbb{R}^n$ with non-empty interior $Ω$, that are attractors of iterated function systems (IFSs) of contracting similarities satisfying the open set condition. Examples for $n=2$ are the closures of the Koch snowflake domain and the Gosper island domain. Our first result is that $Ω$ is thick in the sense of Triebel. A consequence is that $C_0^\infty(Ω)$ is dense in the Sobolev space $H^s_Γ:= \{φ\in H^s(\mathbb{R}^n): \mathrm{supp}(φ)\subset Γ\}$ for all $s\in\mathbb{R}$. Our second result, accompanied by results on pointwise multiplication by characteristic functions and uniform extension operators, is that the spaces $\{H^s(Ω)\}_{s\in \mathbb{R}}$, where $H^s(Ω):=\{φ|_Ω: u\in H^s(\mathbb{R}^n)\}$, form an interpolation scale. This is established as a special case of new extension and interpolation results for Besov and Triebel-Lizorkin spaces, applying to large classes of domains $Ω$ that are thick and have boundary with Assouad dimension $
