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

Journal article

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.

Authors: Ball, J; Capdeboscq, Y; Xiao, B

• Publication status: Published
• Journal: Math. Models Methods Appl. Sci.
• Volume: 22
• Issue: 11
• Pages: 1250036
• Publication date: 2012-01-10
• DOI: 10.1142/S0218202512500364
• EISSN: 1793-6314
• ISSN: 0218-2025
• Language: English
• Keywords: math.AP