Properties of IFS attractors with non-empty interiors, related rough domains, and associated function spaces and scattering problems

TL;DR

This work develops a comprehensive function-space framework for fractal and rough domains arising as interior-containing IFS attractors (n-attractors). It establishes Triebel–Lizorkin/Besov extension and density results, showing that interior/exterior regions are thick in the Triebel sense and belong to the D^t class, which yields multiplier properties and common extension operators across a range of smoothness. The authors derive interpolation-scale properties for Sobolev and generalized Besov/Triebel–Lizorkin spaces on rough domains, and they obtain precise best-approximation rates for piecewise-constant approximations on fractal meshes, with direct application to sound-soft scattering by fractal screens and to convergence analysis of fractal-mesh boundary element methods. The results provide a unified analytical and numerical toolkit for PDEs and boundary-integral equations on fractal and rough boundaries, with potential impact on simulations of wave propagation in complex geometries.

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 $<n$. Our third contribution is to prove best approximation error estimates in fractional negative-order Sobolev spaces for piecewise constant approximations on a ``fractal mesh'' of $Γ$, generated by the IFS, in which the mesh elements are self-similar copies of $Γ$. As an application we study sound-soft acoustic scattering in $\mathbb{R}^{n+1}$ by the fractal screen $Γ\times \{0\}$. Using our density result we prove that the standard PDE formulation of this problem is equivalent to the standard first kind boundary integral equation in which the boundary condition is imposed by restriction to the (relative) interior of the screen. To solve this equation we consider a piecewise-constant Galerkin boundary element method on a fractal mesh, and, using our best approximation error estimates, we prove convergence rates for the Galerkin approximation.

Figures (2)

• Figure 1: Examples of $n$-attractors, $\Gamma\subset\mathbb{R}^n$, in the case $n=2$. In each case the left-hand plot is $\Gamma$, the attractor of some IFS $\{s_1,\ldots,s_M\}$. The right-hand plot is a copy of $\Gamma$ with the subsets $s_1(\Gamma), \ldots s_M(\Gamma)$ numbered, and each shaded in a different colour. All except (d) and (e) are homogeneous, and hence examples of $M$-rep tiles in the sense of bandt1991self.
• Figure 2: The three cases considered in the proof of the $n$-set property in Lemma \ref{['lem:geom']}.

Theorems & Definitions (95)

• Definition 2.1: "$n$-attractor"
• Remark 2.2: Alternative characterisation of $n$-attractors
• Remark 2.3
• Remark 2.4: Examples of $n$-attractors
• Proposition 2.5
• proof
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• proof