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 e are symmetric positive definite matrix-valued functions in ℝ3. We show that a unique continuation result for globally W1, ∞ coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise W1, ∞ with respect to a suitable countable collection of subdomains with C0 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.