Investigation of some transmission problems with sign-changing coefficients. Application to metamaterials.Event type: Thesis defence (Lucas Chesnel)
Start at: october 12, 2012
In this thesis, we study various operators that present a sign-changing in their principal part. These operators appear in particular in electromagnetism when studying wave propagation in structures made of usual materials and negative materials. Here, we say that a material is negative when it is modeled by a dielectric permittivity and/or a magnetic permeability that take(s) negative values. Due to the change of sign of the phy- sical parameters, we can not use the classical tools to study such problems.
In the first part of the thesis, we focus on the scalar transmission problem which is obtained from Maxwells equations when the geometry and the data are invariant in one direction. Using the T-coercivity technique, based on geometric arguments, we establish necessary and sufficient conditions to prove well-posedness for this problem in a bounded domain in H^1. We also show how this approach can be used to justify the convergence of the usual finite element method to approximate the solution.
In a second step, using different techniques coming from the study of elliptic equations in domains with singular geometry, we define a new functional framework to recover Fredholmness when it is lost in H^1. This leads to a surprising black hole phenomenon. Everything happens like if some waves were sucked into a point. We then perform an asymptotic analysis with respect to a small perturbation of the interface between the positive material and the negative material in this functional framework. In our analysis, we observe a curious phenomenon of blinking eigenvalues.
The third part of this thesis is devoted to the study of Maxwells equations. We first work on Maxwells equations in 2D using the results obtained for the scalar problem. Then, we proceed with the 3D Maxwells equations. We show that they are well-posed as long as the associated scalar problems are well-posed.
Finally, in the fourth section, we investigate the interior transmission problem that arises in scattering theory. The operator for this problem also presents a sign-changing in its principal part, and therefore can be studied relying on the analogy with the transmission problem between a positive material and a negative material. For this interior transmission problem, some configurations lead to consider a fourth-order transmission problem with sign-changing coefficients. Our analysis shows that this latter operator has strikingly different properties from those of the second order scalar operator.