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 μ and the permittivity ε are symmetric positive definite matrix-valued functions in 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 subdomains 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 subdomains the permeability and permittivity are only L ∞ in sets of small measure.