Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle

TL;DR

This paper analyzes the shape derivatives of boundary integral operators in electromagnetic scattering and applies this to understand how solutions change with obstacle shape, focusing on dielectric materials.

Contribution

It provides a detailed mathematical analysis of shape differentiability of electromagnetic boundary integral operators and characterizes the first shape derivative of scattering solutions.

Findings

Boundary integral operators are infinitely differentiable without regularity loss.

Shape derivatives can be characterized via new electromagnetic scattering problems.

Analysis is based on Helmholtz decomposition and pseudo-differential operators.

Abstract

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. The latter are typically bounded on the space of tangential vector fields of mixed regularity $TH\sp{-1/2}(\Div_{\Gamma},\Gamma)$. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the G\^ateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.