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

TL;DR

This paper investigates the shape differentiability of boundary integral operators with pseudo-homogeneous kernels in electromagnetic scattering, establishing their infinite differentiability and regularity properties under boundary deformations.

Contribution

It demonstrates the infinite shape differentiability of boundary integral operators and potentials with pseudo-homogeneous kernels, providing a foundation for analyzing electromagnetic scattering problems.

Findings

Boundary integral operators are infinitely differentiable without regularity loss.

Potential operators are infinitely shape differentiable away from the boundary.

Shape differentiability of surface differential operators is also established.

Abstract

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