## Abstract

Let U be a connected, closed, bounded region in ℝ^{3} with smooth boundary 𝛛U. 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 𝛛U, 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) |
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Pages (from-to) | 2196-2230 |

Number of pages | 35 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 72 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2019 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics