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 frequency-independent boundary element method for scattering by two-dimensional screens and apertures

We propose and analyse a hybrid numerical-asymptotic $hp$ boundary element method for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high frequency asymptotics of the solution. Our numerical results suggest that fi�xed accuracy can be achieved at arbitrarily high frequencies with a frequency-independent computational cost. Our analysis does not capture this observed behaviour completely, but we provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom $N$ increases, and that to achieve any desired accuracy it is sufficient to increase $N$ in proportion to the square of the logarithm of the frequency as the frequency increases (standard boundary element methods require $N$ to increase at least linearly with frequency to retain accuracy). We also show how our method can be applied to the complementary "breakwater" problem of propagation through an aperture in an infinite sound-hard screen

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

Cold endoscopic mucosal resection of large sessile serrated polyps at colonoscopy (with video)

The optimal technique for the resection of sessile serrated polyps (SSPs) is unknown, with established limitations and risks with conventional polypectomy. Although cold snare polypectomy is safe, the efficacy of piecemeal resection for large lesions is untested. In this study, we evaluate the safety and efficacy of cold endoscopic mucosal resection (EMR) for large SSPs.Patients presenting for elective colonoscopy at an academic endoscopy center with 1 or more SSPs ≥10mm in size were enrolled, excluding those on anticoagulants or antiplatelets other than aspirin. Lesions were resected with a cold EMR technique comprising submucosal injection of succinylated gelatin and dilute methylene blue before piecemeal cold snare resection of all visible polyp with a margin of normal tissue. Outcomes were the presence of residual serrated neoplasia in biopsies from the defect margin, and findings on surveillance colonoscopy.Cold EMR was performed on 163 SSPs during 105 procedures in 99 patients (97% female; median age 57 years). The mean size was 17.5 mm: 61 SSPs were ≥20 mm, 13 SSPs ≥30 mm, and 97.5% were in the proximal colon. Cytological dysplasia was present in 2 (1.2%). Margin biopsies were positive in 2 (1.2%) lesions. Surveillance colonoscopy for 82% of lesions (median 5 months) showed residual serrated tissue in 1, treated with cold snare, but no evidence of recurrence in the remainder. Minor adverse events were seen in 3 patients; no delayed bleeding was observed.Cold EMR is a safe and effective method for the removal of large SSPs

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