Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators

摘要

In this paper we study the shape differentiability properties of a class ofboundary integral operators and of potentials with weakly singularpseudo-homogeneous kernels acting between classical Sobolev spaces, withrespect to smooth deformations of the boundary. We prove that the boundaryintegral operators are infinitely differentiable without loss of regularity.The potential operators are infinitely shape differentiable away from theboundary, whereas their derivatives lose regularity near the boundary. We studythe shape differentiability of surface differential operators. The shapedifferentiability properties of the usual strongly singular or hypersingularboundary integral operators of interest in acoustic, elastodynamic orelectromagnetic potential theory can then be established by expressing them interms of integral operators with weakly singular kernels and of surfacedifferential operators.