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Abstract
We prove new, sharp, wavenumberexplicit bounds on the norms of the Helmholtz single and doublelayer boundaryintegral 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 secondkind boundaryintegral formulations, posed in L ^{2}(∂Ω) , of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumberexplicit L ^{2}(∂Ω) → H ^{1}(∂Ω) bounds can then be used in a wavenumberexplicit version of the classic compactperturbation analysis of Galerkin discretisations of these secondkind equations; this is done in the companion paper (Galkowski, Müller, and Spence in Wavenumberexplicit analysis for the Helmholtz hBEM: error estimates and iteration counts for the Dirichlet problem, 2017. arXiv:1608.01035).
Original language  English 

Article number  6 
Pages (fromto)  135 
Number of pages  35 
Journal  Integral Equations and Operator Theory 
Volume  91 
Issue number  1 
Early online date  31 Jan 2019 
DOIs  
Publication status  Published  1 Feb 2019 
Keywords
 math.AP
 31B10, 31B25, 35J05, 35J25, 65R20
 Boundary integral equation
 Helmholtz equation
 Semiclassical
 Layerpotential operators
 High frequency
ASJC Scopus subject areas
 Analysis
 Algebra and Number Theory
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Dive into the research topics of 'Wavenumberexplicit regularity estimates on the acoustic single and doublelayer operators'. Together they form a unique fingerprint.Projects
 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council