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    On Generalized Solutions to Some Problems in Electromagnetism and Geometric Optics

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    Genre
    Thesis/Dissertation
    Date
    2016
    Author
    Stachura, Eric Christopher
    Advisor
    Gutiérrez, Cristian E., 1950-
    Committee member
    Berhanu, Shiferaw
    Mendoza, Gerardo A.
    Strain, Robert M.
    Department
    Mathematics
    Subject
    Mathematics
    Electromagnetics
    Optics
    Anisotropic Maxwell Equations
    Inverse Problem
    Maxwell Semigroup
    Monge-ampere Equation
    Negative Refraction
    Weak Solutions
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/2452
    
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    DOI
    http://dx.doi.org/10.34944/dspace/2434
    Abstract
    The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\`ere type arises here.
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