Electromagnetic Boundary Problems

C    Rellich’s Theorem

In this Appendix we will prove a result known as Rellich’s theorem that is an important part of the demonstration of uniqueness for time-harmonic electromagnetic fields. Rellich’s theorem is stated as follows. Let E and H be solutions of Maxwell’s equations with no impressed sources for all r greater than some R₀ in a homogeneous medium whose material parameters are µ and ϵ. The field is not assumed to obey a radiation condition (Rellich’s theorem is used in the proof of a uniqueness theorem that invokes the radiation condition, and we wish to avoid circular arguments). Then if

$\underset{R\to \infty }{\lim }\frac{1}{2}{\displaystyle {\oint }_{S_R}\left[\sqrt{\frac{ϵ}{\mu }}{\left|{\mathbf{E}}_{\tan }\right|}^2+\sqrt{\frac{\mu }{ϵ}}{\left|{\mathbf{H}}_{\tan }\right|}^2\right]dS=0}$

(C.1)

where S_R is a sphere of radius R centered at the origin, then E and H must vanish identically ...