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Stable Skeleton Integral Equations for General-Coefficient Helmholtz Transmission Problems

by B. Gräßle and R. Hiptmair and S. Sauter

(Report number 2025-20)

Abstract
A novel variational formulation of layer potentials and boundary integral op- erators generalizes their classical construction by Green’s functions, which are not explicitly available for Helmholtz problems with variable coefficients. Wavenumber explicit estimates and properties like jump conditions follow directly from their variational definition and enable a non-local (“integral”) formulation of acoustic transmission problems (TP) with piecewise Lipschitz coefficients. We obtain the well-posedness of the integral equations directly from the stability of the underlying TP. The simultaneous analysis for general dimensions and complex wavenumbers (in this paper) imposes an artificial boundary on the external Helmholtz problem and employs recent insights into the associated Dirichlet-to-Neumann map.

Keywords: acoustic wave propagation, variable coefficients, transmission problem, layer potential, single-trace formulation, multi-trace formulation

@Techreport{GHS25_1141,
  author = {B. Gr\"aßle and R. Hiptmair and S. Sauter},
  title = {Stable Skeleton Integral Equations for General-Coefficient Helmholtz Transmission Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-20.pdf },
  year = {2025}
}

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