Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains

TL;DR

This paper extends the mathematical framework for Helmholtz transmission and impedance scattering problems to complex, irregular, and fractal boundaries, broadening the scope of analysis and potential applications in acoustic scattering.

Contribution

It generalizes layer potential, Neumann-Poincaré operators, and Calderón projectors to extension domains with irregular boundaries, enabling analysis of scattering problems without boundary measure constraints.

Findings

Established well-posedness of impedance scattering in generalized boundary spaces

Connected transmission problems to one-sided scattering problems with various boundary conditions

Extended optimization results for acoustic scattering to complex domain boundaries

Abstract

We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{\'e} operators, and of Calder{\'o}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension.