The Debye source representation for solutions to the time-harmonic Maxwell equations is extended to bounded domains with finitely many smooth boundary components. A strong uniqueness result is proved for this representation. Natural complex structures are identified on the vector spaces of time-harmonic Maxwell fields. It is shown that these complex structures are uniformized by the Debye source representation, that is, represented by a fixed linear map on a fixed vector space, independent of the frequency. This complex structure relates time-harmonic Maxwell fields to constant-k Beltrami fields, i.e., solutions of the equation ∇×E=kE. A family of self-adjoint boundary conditions are defined for the Beltrami operator. This leads to a proof of the existence of zero-flux, constant-k, force-free Beltrami fields for any bounded region in R3, as well as a constructive method to find them. The family of self-adjoint boundary value problems defines a new spectral invariant for bounded domains in R3.
ASJC Scopus subject areas
- Applied Mathematics