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

Abstract

We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability $\mu$ and the permittivity $\epsilon$ 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.

9 pages, 4 figures

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