On potential-based shape derivatives of the electromagnetic transmission problem

Domain derivatives are an important tool to characterize and compute shape derivatives. If some quantity of interest depends on the shape of an object such as the obstacle in a scattering problem, shape derivatives are used to describe the effect of variations of the shape on that quantity. We here consider the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. As an alternative to the formulation using the Maxwell system, the problem may be posed as a coupled system of Helmholtz equations with complicated transmission conditions. We prove equivalence of the two formulations and then proceed to characterize the domain derivatives of the scattered fields in the potential formulation. Our main result is the equivalence of the characterizations of such derivatives in the Maxwell and in the potential based problem formulation.