## Abstract

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 language | English |
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Article number | 36 |

Journal | Integral Equations and Operator Theory |

Volume | 94 |

Issue number | 4 |

Early online date | 17 Oct 2022 |

DOIs | |

Publication status | E-pub ahead of print - 17 Oct 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory