Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, or (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-to- Dirichlet map when condition (i) and (ii) hold. We also prove bounds on the solutions of the interior and exterior impedance problems when the obstacle is a general Lipschitz domain. These bounds are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber. One motivation for proving these collection of bounds is that they can then be used to prove wavenumber-explicit bounds on the inverses of the standard second-kind integral operators used to solve the exterior Dirichlet, Neumann, and impedance problems for the Helmholtz equation.

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