Inverse Problems for Partial Differential Equations

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The proposed work has four major goals. (1) The first goal is to further advance the work concerning cloaking-by-mapping. Particular emphasis will be placed on the physical relevance of the proposed meta-materials (used for the cloak), and on the potential for improvement in the relative cloaking rates of the approximate schemes. In the context of physical relevance, the goal is to study meta- materials that satify the socalled Kramers-Kronig relations (which naturally emerge from causality considerations). Initially these studies will focus on materials, which are governed by the socalled Drude-Lorentz model. It is also planned to perform a careful study of the similarities between the technique of perfectly matched layers (PML) and cloaking-by-mapping. (2) The second goal is to better estimate the presence of inhomogeneities, in particular their volume fraction, from the response to multiple boundary loads. The aim is to extend earlier low volume fraction results, obtained for small number of boundary measurements (n measurements in n dimensions). (3) The third goal is to study the uniqueness and stability of an inverse problem, in which one seeks to identify a (nonlinear) current flux from a single (Neumann) boundary measurement. The associated boundary value problem is motivated by the equations of magnetohydrodynamic equilibria. (4) The fourth and final goal is to study static and dynamic problems associated with corrosion and oxidation modelling. These studies will range from the construction of explicit solutions, to the study of boundary blow-up, and criteria for finite time blow-up vs. existence for all time.There are significant possibilities for broader impact. In the area of "cloaking-by-mapping", the kind of results, one can expect to obtain, will be extremely useful in determining the degree of "near invisibility" that is in practice achievable, given (physically relevant) limitations on the meta-materials, used to construct the cloaks. One can well imagine an interplay between the type of results expected, and the actual physical construction of approximate invisibility cloaks. The work on the estimation/detection of small inhomogeneities has already led to the development of algorithms of use in flaw detection and medical imaging. The improved bounds, for volume fractions, will have similar potential applications. While the studies of the identifiability of the electric flux in the context of magnetohydrodynamics will be mostly theoretical, it is clear that the study of stability properties, and of the benefits of using additional line-integral measurements, could have very practical applications in terms of the control of plasmas in Tokamak devices.