Acoustic scattering by fractal screens: mathematical formulations and wavenumber-explicit continuity and coercivity estimates

We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. In contrast to previous studies, in which the domain occupied by the screen is assumed to be Lipschitz or smoother, we consider screens occupying an arbitrary bounded open set in the plane. Thus our study includes cases where the closure of the domain occupied by the screen has larger planar Lebesgue measure than the screen, as can happen, for example, when the screen has a fractal boundary. We show how to formulate well-posed boundary value problems for such scattering problems, our arguments depending on results on the coercivity of the acoustic single-layer and hypersingular boundary integral operators, and on properties of Sobolev spaces on general open sets which appear to be new. Our analysis teases out the explicit wavenumber dependence of the continuity and coercivity constants of the boundary integral operators, viewed as mappings between fractional Sobolev spaces, this in part extending previous results of Ha-Duong. We also consider the complementary problem of propagation through a bounded aperture in an infinite planar screen.