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

Sylvain Cappell, Edward Miller

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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 languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StatePublished - Jan 1 2019

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Maxwell equations
Maxwell's equations
Positivity
Positive definite
Magnetic permeability
Harmonic
Charge density
Tensors
Manifolds with Boundary
Hermitian matrix
Permittivity
Current density
Permeability
Waveguide
Riemannian Manifold
Tensor
Charge
Oscillation
Necessary Conditions
Closed

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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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.",
author = "Sylvain Cappell and Edward Miller",
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AU - Miller, Edward

PY - 2019/1/1

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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.

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