### Abstract

Let U be a connected, closed, bounded region in ℝ
^{3}
with smooth boundary 훛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e
^{−iωt}
. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛ
_{ω}
and magnetic permeability μ
_{ω}
, which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, E
_{T}
or H
_{T}
on 훛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified E
_{T}
, H
_{T}
, respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric E
_{T}
case, the positivity condition is this: the complex Hermitian matrix μ
_{ω}
(p) is to be positive definite while only the real part of ɛ
_{ω}
(p), i.e., Re(ɛ
_{ω}
(p)), necessarily real symmetric, need be positive definite. In the magnetic-type H
_{T}
case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides.

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

Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region.** / Cappell, Sylvain; Miller, Edward.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region

AU - Cappell, Sylvain

AU - Miller, Edward

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let U be a connected, closed, bounded region in ℝ 3 with smooth boundary 훛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e −iωt . We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛ ω and magnetic permeability μ ω , which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, E T or H T on 훛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified E T , H T , respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric E T case, the positivity condition is this: the complex Hermitian matrix μ ω (p) is to be positive definite while only the real part of ɛ ω (p), i.e., Re(ɛ ω (p)), necessarily real symmetric, need be positive definite. In the magnetic-type H T case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides.

AB - Let U be a connected, closed, bounded region in ℝ 3 with smooth boundary 훛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e −iωt . We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛ ω and magnetic permeability μ ω , which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, E T or H T on 훛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified E T , H T , respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric E T case, the positivity condition is this: the complex Hermitian matrix μ ω (p) is to be positive definite while only the real part of ɛ ω (p), i.e., Re(ɛ ω (p)), necessarily real symmetric, need be positive definite. In the magnetic-type H T case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides.

UR - http://www.scopus.com/inward/record.url?scp=85065796192&partnerID=8YFLogxK

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U2 - 10.1002/cpa.21821

DO - 10.1002/cpa.21821

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -