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

  • Jeffrey Galkowski
  • Euan A. SpenceEmail author
Open Access
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Abstract

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from \({L^2({\partial {\Omega }})}\rightarrow H^1({\partial {\Omega }})\) (where \({\partial {\Omega }}\) 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 \({L^2({\partial {\Omega }})}\), of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit \({L^2({\partial {\Omega }})}\rightarrow H^1({\partial {\Omega }})\) 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).

Keywords

Helmholtz equation Layer-potential operators High frequency Semiclassical Boundary integral equation 

Mathematics Subject Classification

31B10 31B25 35J05 35J25 65R20 

Notes

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Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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