Stable skeleton integral equations for general coefficient Helmholtz transmission problems

TL;DR

This paper introduces a new variational approach to boundary integral equations for Helmholtz transmission problems with variable coefficients, enabling stable and well-posed formulations across dimensions and complex wavenumbers.

Contribution

It develops a generalized variational formulation of layer potentials that does not rely on explicit Green's functions, facilitating analysis of variable coefficient Helmholtz problems.

Findings

Establishes well-posedness of the integral equations.

Provides wavenumber explicit estimates and jump conditions.

Handles general dimensions and complex wavenumbers.

Abstract

A novel variational formulation of layer potentials and boundary integral operators 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.