Nonlocal impedance conditions in direct and inverse obstacle scattering

We discuss the use of nonlocal impedance conditions within the use of boundary integral equations for the solution of direct and inverse obstacle scattering problems for penetrable obstacles with constant index of refraction. In the first part, for the classical transmission problem we present an approach that leads to a two-by-two system of nonlinear integral equations in the spirit of the method initiated by Kress and Rundell (2005 Inverse Problems 21 1207-23) rather than the three-by-three system arising from the traditional boundary integral equation approach to the transmission conditions. In the second part we survey on recent work of Cakoni and Kress (2017 Appl. Anal. 96 23-38) on the use of boundary integral equations for the characterization and numerical computation of transmission eigenvalues. In particular, we modify and simplify the analysis by the use of a nonlocal rather than a local impedance condition as in the 2017 paper.