Dirichlet-to-Neumann operator for the Helmholtz problem with general wavenumbers on the $n$-sphere
Abstract
This paper considers the Helmholtz problem in the exterior of a ball with Dirichlet boundary conditions and radiation conditions imposed at infinity. The differential Helmholtz operator depends on the complex wavenumber with non-negative real part and is formulated for general spatial dimension. We prove wavenumber explicit continuity estimates of the corresponding Dirichlet-to-Neumann (DtN) operator which are valid for all wavenumbers under consideration and do not deteriorate as they tend to zero. The exterior Helmholtz problem can be equivalently reformulated on a bounded domain with DtN boundary conditions on the artificial boundary of a ball. We derive wavenumber independent trace and Friedrichs-type inequalities for the solution space in wavenumber-indexed norms.
Keywords
Cite
@article{arxiv.2503.18837,
title = {Dirichlet-to-Neumann operator for the Helmholtz problem with general wavenumbers on the $n$-sphere},
author = {Benedikt Gräßle and Stefan A. Sauter},
journal= {arXiv preprint arXiv:2503.18837},
year = {2025}
}