Wavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screens

TL;DR

This paper establishes wavenumber-explicit continuity and coercivity estimates for boundary integral operators in acoustic scattering by planar screens, enhancing understanding of their mathematical properties in fractional Sobolev spaces.

Contribution

It provides the first explicit bounds on continuity and coercivity constants for these operators, based on spectral analysis, advancing theoretical understanding of acoustic scattering models.

Findings

Proves continuity of boundary integral operators with wavenumber dependence.

Establishes coercivity estimates explicitly involving the wavenumber.

Uses spectral representations to derive bounds in fractional Sobolev spaces.

Abstract

We study the classical first-kind boundary integral equation reformulations of time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. We prove continuity and coercivity of the relevant boundary integral operators (the acoustic single-layer and hypersingular operators respectively) in appropriate fractional Sobolev spaces, with wavenumber-explicit bounds on the continuity and coercivity constants. Our analysis is based on spectral representations for the boundary integral operators, and builds on results of Ha-Duong (Jpn J Ind Appl Math 7:489--513 (1990) and Integr Equat Oper Th 15:427--453 (1992)).