Geometric Structures of Maxwellian Eigenfunctions

Huaian Diao and Hongyu Liu
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Huaian Diao: Jilin University, School of Mathematics
Hongyu Liu: City University of Hong Kong, Department of Mathematics

Chapter Chapter 3 in Spectral Geometry and Inverse Scattering Theory, 2023, pp 63-89 from Springer

Abstract: Abstract We are concerned with the unique continuation property (UCP) of the time-harmonic Maxwell system ∇ ∧ E − i k H = 0 , ∇ ∧ H + i k E = 0 in Ω , $$\displaystyle \nabla \wedge \mathbf {E}-\mathrm {i} k\mathbf {H}={\mathbf 0}\,, \quad \nabla \wedge \mathbf {H}+\mathrm {i} k\mathbf {E}={\mathbf 0} \quad \mbox{in} ~~\varOmega , $$ where i : = − 1 $$\mathrm {i}:=\sqrt {-1}$$ and k ∈ ℝ + $$k\in \mathbb {R}_+$$ , in a particular scenario that is strongly motivated by our study of the inverse electromagnetic scattering problem. In this chapter, we follow the treatment in Diao et al. (Inverse Probl 37:035004, 2021). Further discussion on the inverse electromagnetic scattering problems can be found in Sect. 4.1.2 . We start with the necessary mathematical setup. We consider the domain Ω to be an open set in ℝ 3 $$\mathbb {R}^3$$ , bounded or unbounded, and the solution (E, H) to the system (3.1.1) in the space Hloc(curl, Ω) defined by H l o c ( curl , Ω ) = { U | B ∈ H ( curl , B ) ; B is any bounded subdomain of Ω } , H ( curl , B ) = { U ∈ L 2 ( B ) 3 ; ∇ ∧ U ∈ L 2 ( B ) 3 } . $$\displaystyle \begin {array}{rl} H_{loc}(\mathrm {curl}, \varOmega )=& \big \{U|{ }_{B}\in H(\mathrm {curl}, B); B\ \mbox{is any bounded subdomain of}\ \varOmega \big \}, \\ H(\mathrm {curl}, B)=& \big \{U\in L^2(B)^3; \ \nabla \wedge U\in L^2(B)^3\big \}. \end {array} $$

Date: 2023
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DOI: 10.1007/978-3-031-34615-6_3

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