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Inverse problems for the anisotropic Maxwell equations. (English) Zbl 1226.35086

Let \((M,g)\) be a compact, 3-dimensional Riemannian manifold with boundary \(\partial M\). The time-harmonic Maxwell equations read \(*dE=i\omega\mu H\) and \(*dH=-i\omega\varepsilon E\), where \(d\) denotes the exterior derivative, and \(*\) is the Hodge star operator. The inverse problem, at a fixed non-resonant frequency \(\omega>0\), consists of recovering the material parameters \(\varepsilon\) and \(\mu\) from the admittance map \(\Lambda:tE\mapsto tH\), where \(t\) is the tangential trace on forms. In Theorem 1.1, assuming admissability of the metric, uniqueness of the inverse problem is proved for complex-valued material parameters. In Theorem 1.2, which is essentially a corollary to Theorem 1.1, uniqueness of the inverse problem is shown for a Euclidean domain if the material parameters are positive definite \((1,1)\)-tensors, which belong to the conformal class of an admissable metric. The authors assert that these theorems are the first positive results on the inverse problem for time-harmonic Maxwell equations in anisotropic settings.
The admissability assumption on metrics was introduced by D. Dos Santos Ferreira and the authors in [Invent. Math. 178, No. 1, 119–171 (2009; Zbl 1181.35327)]. It is fundamental to a construction of complex geometric optics solutions. In the paper under review, unique complex geometric optics solutions for the Maxwell system are obtained by reducing to the Schrödinger equation with Hodge Laplacian. These solutions are then inserted into an integral identity, leading to the recovery of the electromagnetic parameters.

MSC:

35R30 Inverse problems for PDEs
35Q61 Maxwell equations

Citations:

Zbl 1181.35327
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References:

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