We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.
In weakly coupled, current biased, doped semiconductor superlattices, domain walls may move upstream against the flow of electrons. For appropriate doping values, a domain wall separating two electric-field domains moves downstream below a first critical current, it remains stationary between this value and a second critical current, and then moves upstream above. These conclusions are reached by using a comparison principle to analyze a discrete drift-diffusion model, and validated by numerical simulations. Possible experimental realizations are suggested.
We study the stability and evolution of various elastic defects in a flat graphene sheet and the electronic properties of the most stable configurations. Two types of dislocations are found to be stable: "glide" dislocations consisting of heptagon-pentagon pairs, and "shuffle" dislocations, an octagon with a dangling bond. Unlike the most studied case of carbon nanotubes, Stone Wales defects are unstable in the planar graphene sheet. Similar defects in which one of the pentagonheptagon pairs is displaced vertically with respect to the other one are found to be dynamically stable. Shuffle dislocations will give rise to local magnetic moments that can provide an alternative route to magnetism in graphene.
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