On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body

TL;DR

This paper develops and analyzes single integral equation methods for electromagnetic scattering by dielectric bodies, ensuring unique solvability across all real frequencies by appropriate parameter choices.

Contribution

It introduces four families of integral equations for dielectric scattering that are uniquely solvable for all real frequencies, extending Kleinman-Martin ideas to electromagnetics.

Findings

Four families of integral equations are shown to be uniquely solvable for all real frequencies.

The well-posedness of these equations is established on smooth and non-smooth boundaries.

The methods improve the robustness of electromagnetic scattering solutions.

Abstract

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface $\Gamma$. In this paper, following an idea developed by Kleinman and Martin \cite{KlMa} for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over $\Gamma$ for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the well-posedness of the integral equations in the space of finite energy on smooth and non-smooth boundaries.