Mathematical basis of the linear sampling method for partially coated obstacles

We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method.