Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains

Abstract

We consider a transmission problem for the Helmholtz equation with a fixed, positive wavenumber across the boundary of an extension domain. Such a boundary can be Lipschitz, fractal, or of varying Hausdorff dimension. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}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, and restate their well-posedness as boundary equations. 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 it 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.