Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields. Issue 12 (30th January 2015)

• Record Type: Journal Article
• Authors: Epstein, Charles L.; Greengard, Leslie; O'Neil, Michael
• Is Part Of: Communications on pure and applied mathematics. Volume 68:Issue 12(2015:Dec.)
• Page Start: 2237
• Page End: 2280
• Publication Date: 2015-01-30
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21560
• Languages: English
• ISSNs: 0010-3640

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>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‐<italic>k</italic> Beltrami fields, i.e., solutions of the equation <disp-formula content-type="mathematics" id="cpa21560-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrn9n1sb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:00103640:media:cpa21560:cpa21560-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∇</mml:mo><mml:mo>×</mml:mo><mml:mi mathvariant="bold">E</mml:mi><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo> </mml:mo><mml:mi mathvariant="bold">E</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula> 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‐<italic>k</italic>, force‐free Beltrami fields for any bounded region in ℝ<sup>3</sup>, 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<sup>3</sup>.© 2015 Wiley Periodicals, Inc.</p> </abstract>