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

TL;DR

This paper establishes the uniqueness of solutions to time-harmonic anisotropic Maxwell's equations with piecewise regular coefficients, extending known results to less regular and piecewise-defined material parameters.

Contribution

It proves a new uniqueness result for Maxwell's equations with piecewise $W^{1, ext{infty}}$ and $L^ ext{infty}$ coefficients, broadening the class of admissible material properties.

Findings

Uniqueness holds for solutions with piecewise $W^{1, ext{infty}}$ coefficients.

Extension to coefficients that are only $L^ ext{infty}$ in small measure sets.

General argument applicable beyond Maxwell's equations.

Abstract

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.