Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators

• Published in: Integral equations and operator theory 2019-02, Vol.91 (1), p.1-35, Article 6
• Main Authors: Galkowski, Jeffrey, Spence, Euan A.
• Format: Article
• Language: English
• Subjects: Analysis; Dirichlet problem; Error analysis; Estimates; Formulations; Galerkin method; Helmholtz equations; Integrals; Mathematics; Mathematics and Statistics; Norms; Operators (mathematics); Perturbation methods; Wavelengths
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-019-2502-x

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from L 2 ( ∂ Ω ) → H 1 ( ∂ Ω ) (where ∂ Ω is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in L 2 ( ∂ Ω ) , of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit L 2 ( ∂ Ω ) → H 1 ( ∂ Ω ) bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper (Galkowski, Müller, and Spence in Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem, 2017 . arXiv:1608.01035 ).