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

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

Research output: Contribution to journalArticle

Abstract

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
Volume68
Issue number12
DOIs
StatePublished - Dec 1 2015

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Vector spaces
Complex Structure
Harmonic
Maxwell equations
Boundary value problems
Vector space
Mathematical operators
Bounded Domain
Boundary conditions
Fluxes
Linear map
Maxwell's equations
Uniqueness
Boundary Value Problem
Invariant
Zero
Operator
Family

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields. / Epstein, Charles L.; Greengard, Leslie; O'Neil, Michael.

In: Communications on Pure and Applied Mathematics, Vol. 68, No. 12, 01.12.2015, p. 2237-2280.

Research output: Contribution to journalArticle

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