### 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 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}.

Original language | English (US) |
---|---|

Pages (from-to) | 2237-2280 |

Number of pages | 44 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 68 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*68*(12), 2237-2280. https://doi.org/10.1002/cpa.21560

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

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 68, no. 12, pp. 2237-2280. https://doi.org/10.1002/cpa.21560

}

TY - JOUR

T1 - Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields

AU - Epstein, Charles L.

AU - Greengard, Leslie

AU - O'Neil, Michael

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84945187891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945187891&partnerID=8YFLogxK

U2 - 10.1002/cpa.21560

DO - 10.1002/cpa.21560

M3 - Article

VL - 68

SP - 2237

EP - 2280

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -