Mathematical Sciences: Mathematical Problems in Electromagnetism and Elasticity

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This project involves mathematical modeling of electromagnetic and elastic materials. These materials are characterized by equilibrium configurations exhibiting multiple phases. Examples include elastic ferromagnets and rigid superconductors. The basic modeling tools will be the use of spatially nonlocal energy terms to penalize oscillations in classical states and the variance of Young-measure states. Also included are existence results for short time dynamics of materials satisfying a generalized strong ellipticity hypothesis on electromagnetic and elastic properties and existence of equilibrium solutions for elastic bodies with perfect diamagnetism (elastic superconductors.) These studies will lead to a better understanding of the mechanical properties of elastic and electromagnetic materials and composites and will have an impact on the development and synthesis of materials with optimal properties.