High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

TL;DR

This paper establishes new high-frequency bounds on the boundary integral operators used in solving the Helmholtz exterior Neumann problem, providing sharp estimates on their norms and condition numbers for various obstacle geometries.

Contribution

It provides the first frequency-explicit bounds on the condition number of boundary integral operators for general obstacles, extending previous results beyond spherical cases.

Findings

Bounds on the operator norm are sharp up to log factors.

Inverse norm bounds are sharp for smooth obstacles with positive curvature.

First proven $L^2$ condition-number bounds for non-spherical obstacles.

Abstract

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $\Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(\Gamma)$ 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 $\Gamma$ 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 $\Gamma$ and are observed to be sharp at least when $\Gamma$ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on $L^2(\Gamma)$; this is the first time $L^2(\Gamma)$ condition-number bounds have been proved for this operator for obstacles other than balls.