On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients - Ball, J; Capdeboscq, Y; Xiao, B (2012)
We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability $μ$ and the permittivity $ε$ are symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We show that a unique continuation result for globally $W^{1,∞}$ coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise $W^{1,∞}$ with respect to a suitable countable collection of sub-domains with $C^{0}$ boundaries. Such suitable collections include any bounded finite collection. The proof relies on a general argument, not specific to Maxwell's equations. This result is then extended to the case when within these sub-domains the permeability and permittivity are only $L^∞$ in sets of small measure.