Electromagnetic wave propagation in media consisting of dispersive metamaterials

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