Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields

Charles L. Epstein, Leslie Greengard, Michael O'Neil

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2237-2280
Number of pages44
JournalCommunications on Pure and Applied Mathematics
Issue number12
StatePublished - Dec 2015

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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