Boundary conditions for Helmholtz reductions of Maxwell’s equations ensuring ellipticity and equivalence

Determine the additional boundary conditions that must be imposed on the Helmholtz equations obtained by applying the curl operator to the time-harmonic Maxwell system in a bounded or unbounded domain Ω ⊂ ℝ^3 so as to guarantee both (i) ellipticity of the resulting boundary value problem and (ii) preservation of equivalence with the original boundary value problem for Maxwell’s equations.

Background

The paper discusses embedding boundary value problems for time-harmonic Maxwell’s equations into elliptic boundary value theory. In the whole space, applying curl reduces the problem to Helmholtz equations, but this approach becomes problematic in domains with boundaries because appropriate boundary conditions for the Helmholtz equations must be specified.

The authors explicitly point out that in bounded or unbounded domains it is unclear which additional boundary conditions should be added to ensure both ellipticity and consistency with the original Maxwell problem. They propose an alternative approach by augmenting Maxwell’s system with two scalar functions and suitable boundary conditions to form an elliptic problem, thereby sidestepping—but not resolving—the question of the correct Helmholtz boundary conditions. The difficulty is even more pronounced for transmission problems with interfaces.