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

Abstract

We study fractal sets $\Gamma\subset \mathbb{R}^n$ with non-empty interior $\Omega$, 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 $\Omega$ is thick in the sense of Triebel. A consequence is that $C_0^\infty(\Omega)$ is dense in the Sobolev space $H^s_\Gamma:= \{\phi\in H^s(\mathbb{R}^n): \mathrm{supp}(\phi)\subset \Gamma\}$ 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 $\{H^s(\Omega)\}_{s\in \mathbb{R}}$, where $H^s(\Omega):=\{\phi|_\Omega: u\in H^s(\mathbb{R}^n)\}$, 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 $\Omega$ 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 $\Gamma$, generated by the IFS, in which the mesh elements are self-similar copies of $\Gamma$. As an application we study sound-soft acoustic scattering in $\mathbb{R}^{n+1}$ by the fractal screen $\Gamma\times \{0\}$. 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.