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

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 $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure and $\Omega$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $\Gamma\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ 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 $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}=\{0\}$ for a given closed set $\Gamma\subset\mathbb R^n$ 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.Comment: 38 pages, 6 figure

A note on properties of the restriction operator on Sobolev spaces

In our companion paper [3] we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset Ω of Rn, defined as closed subspaces of the classical Bessel potential spaces Hs(Rn) for s∈R. These spaces are mapped by the restriction operator to certain spaces of distributions on Ω. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces. We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case

Well-posed PDE and integral equation formulations for scattering by fractal screens

We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens embedded in $\R^n$ for $n=2$ or $3$. In contrast to previous studies in which the screen is assumed to be a bounded Lipschitz (or smoother) relatively open subset of the plane, we consider screens occupying arbitrary bounded subsets. Thus our study includes cases where the screen is a relatively open set with a fractal boundary, and cases where the screen is fractal with empty interior. We elucidate for which screen geometries the classical formulations of screen scattering are well-posed, showing that the classical formulation for sound-hard scattering is not well-posed if the screen boundary has Hausdorff dimension greater than $n-2$. Our main contribution is to propose novel well-posed boundary integral equation and boundary value problem formulations, valid for arbitrary bounded screens. In fact, we show that for sufficiently irregular screens there exist whole families of well-posed formulations, with infinitely many distinct solutions, the distinct formulations distinguished by the sense in which the boundary conditions are understood. To select the physically correct solution we propose limiting geometry principles, taking the limit of solutions for a sequence of more regular screens converging to the screen we are interested in, this a natural procedure for those fractal screens for which there exists a standard sequence of prefractal approximations. We present examples exhibiting interesting physical behaviours, including penetration of waves through screens with "holes" in them, where the "holes" have no interior points, so that the screen and its closure scatter differently. Our results depend on subtle and interesting properties of fractional Sobolev spaces on non-Lipschitz sets

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper [Gibbs, Hewett and Moiola, Numer. Alg., 2023] it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake

Accelerated Calder\'on preconditioning for Maxwell transmission problems

We investigate a range of techniques for the acceleration of Calder\'on (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve significant reductions in computational cost by (i) using "reduced" Calder\'on preconditioners obtained by discarding constituent boundary integral operators that are not essential for regularisation, and (ii) adopting a ``bi-parametric'' approach in which we use a lower quality (cheaper) $\mathcal{H}$-matrix assembly routine for the preconditioner than for the original operator, including a novel approach of discarding far-field interactions in the preconditioner. Using the boundary element software Bempp (www.bempp.com), we compare the performance of different combinations of these techniques in the context of scattering by multiple dielectric particles. Applying our accelerated implementation to 3D electromagnetic scattering by an aggregate consisting of 8 monomer ice crystals of overall diameter 1cm at 664GHz leads to a 99% reduction in memory cost and at least a 75% reduction in total computation time compared to a non-accelerated implementation