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 dimensions. We prove wavenumber explicit continuity estimates of the corresponding Dirichlet-to-Neumann (DtN) operator which do not deteriorate as the complex wavenumber tends 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.