2013/08/25 by Charles L. Epstein, Leslie Greengard, Epstein, Charles L. +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Plasma Physics (physics.plasm-ph)
paper · pdf · doi:10.48550/arxiv.1308.5425
openalex publication_date 2013/08/25 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The Debye source representation for solutions to the time harmonic Maxwell\nequations is extended to bounded domains with finitely many smooth boundary\ncomponents. A strong uniqueness result is proved for this representation.\nNatural complex structures are identified on the vector spaces of time-harmonic\nMaxwell fields. It is shown that in terms of Debye source data, these complex\nstructures are uniformized, that is, represented by a fixed linear map on a\nfixed vector space, independent of the frequency. This complex structure\nrelates time-harmonic Maxwell fields to constant-k Beltrami fields, i.e.\nsolutions of the equation curl(E) = kE. A family of self-adjoint boundary\nconditions are defined for the Beltrami operator. This leads to a proof of the\nexistence of zero-flux, constant-k, force-free Beltrami fields for any bounded\nregion in R3, as well as a constructive method to find them. The family of\nself-adjoint boundary value problems defines a new spectral invariant for\nbounded domains in R3.\n