We establish the well-posedness, the finite speed propagation, and aregularity result for Maxwell's equations in media consisting of dispersive(frequency dependent) metamaterials. Two typical examples for suchmetamaterials are materials obeying Drude's and Lorentz' models. The causalityand the passivity are the two main assumptions and play a crucial role in theanalysis. It is worth noting that by contrast the well-posedness in thefrequency domain is not ensured in general. We also provide some numericalexperiments using the Drude's model to illustrate its dispersive behaviour.
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