Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region

Let U be a connected, closed, bounded region in ℝ³ with smooth boundary 𝛛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e^−iωt. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛ_ω and magnetic permeability μ_ω, which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, E_T or H_T on 𝛛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified E_T, H_T, respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric E_T case, the positivity condition is this: the complex Hermitian matrix μ_ω(p) is to be positive definite while only the real part of ɛ_ω(p), i.e., Re(ɛ_ω(p)), necessarily real symmetric, need be positive definite. In the magnetic-type H_T case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides.