Embedding transmission problems for Maxwell's equations into elliptic theory

TL;DR

This paper transforms boundary value problems for time-harmonic Maxwell's equations into elliptic problems by introducing scalar functions and boundary conditions, enabling new analytical approaches.

Contribution

It introduces a novel method to embed Maxwell's boundary problems into elliptic theory using additional scalar functions and boundary conditions.

Findings

Established a one-to-one correspondence between Maxwell and elliptic solutions.

Extended the approach to bounded and unbounded domains.

Applied the method to transmission problems with inhomogeneities.

Abstract

We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem.