High-Frequency Estimates on Boundary Integral Operators for the Helmholtz Exterior Neumann Problem

Euan Spence, J. Galkowski, Pierre Marchand

Research output: Contribution to journalArticlepeer-review


We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L 2(Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp up to factors of log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on L 2(Γ) ; this is the first time L 2(Γ) condition-number bounds have been proved for this operator for obstacles other than balls.

Original languageEnglish
Article number36
JournalIntegral Equations and Operator Theory
Issue number4
Early online date17 Oct 2022
Publication statusE-pub ahead of print - 17 Oct 2022


  • Boundary integral equation
  • Helmholtz
  • High frequency
  • Neumann problem
  • Pseudodifferential operator
  • Semiclassical analysis

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory


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