Dirichlet-to-Neumann operator for the Helmholtz problem with general wavenumbers on the n-sphere

2025/03/24 by Gräßle, Benedikt, Sauter, Stefan A. · 2 citations
#31B10 #35J05 #47G10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2503.18837

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.

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