On uniqueness for time harmonic anisotropic Maxwell's equations with
piecewise regular coefficients

We are interested in the uniqueness of solutions to Maxwell's equations when
the magnetic permeability $\mu$ and the permittivity $\varepsilon$ are
symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We
show that a unique continuation result for globally $W^{1,\infty}$ coefficients
in a smooth, bounded domain, allows one to prove that the solution is unique in
the case of coefficients which are piecewise $W^{1,\infty}$ 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 $L^\infty$ in sets of small measure.