Shape derivatives of boundary integral operators in electromagnetic scattering

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. To this end, we start with the Gâteaux differentiability analysis with respect to deformations of the obstacle of boundary integral operators with pseudo-homogeneous kernels acting between Sobolev spaces. The boundary integral operators of electromagnetism are typically bounded on the space of tangential vector fields of mixed regularity TH− 1 2 (divΓ,Γ). Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem.