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Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
Author(s) -
Epstein Charles L.,
Greengard Leslie,
O'Neil Michael
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21560
Subject(s) - mathematics , maxwell's equations , bounded function , mathematical analysis , uniqueness , boundary value problem
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 = k E . 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 ℝ 3 , 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 ℝ 3 .© 2015 Wiley Periodicals, Inc.