Variational Problems and their Applications

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This proposal addresses several aspects of multiscale applied analysis within the areas of the Calculus of Variations, Partial Differential Equations, and Geometric Measure Theory. The work will entail relaxation and lower-semicontinuity results, development of higher order theories, multiple scale homogenization of length and time scales, and dimension reduction problems. The questions to be studied by the investigator and her collaborators include: - in micromagnetics, the derivation of a model for large bodies from the small bodies model which exhibits competition between the anisotropic energy and the exchange energy terms, optimal design and multiscale homogenization of poly-magnetic, different grains materials; - in thin films, the identification of the asymptotic energy as the thickness of the sample vanishes as depending on the macroscopic deformation and on the bending moment, and the characterization of Young measures generated by scaled thin film gradients. Instabilities in the heteroepitaxial growth of thin films over substrates (wet and dry wetting), and in particular the formation of islands with zero contact angle, will be studied; - the microphase separation of copolymer melts and the identification via Gamma-convergence techniques of the interfacial energy between pairs of adjacent monomers; - the asymptotic behavior of optimal urban networks as the length of the network goes to infinity, within the Monge-Kantorovich setting; - the stability of foams under the influence of surfactants, and the role of surfactants in the nucleation of Plateau borders. The research program outlined above is strongly motivated by the need to develop innovative applied mathematics capable to respond to challenges in high-end technology. Novel man-made materials often exhibit underlying models at the forefront of traditional mathematical theories. These include shape memory alloys, ferroelectric, electromagnetic and magnetostrictive materials, composites, liquid crystals, foams used in oil recovery, detergents and lightweight structural materials, and thin films. A large range of length and time scales is usually present, and multiscale aspects yield applications from bulk materials to nanostructures. In order to understand and be able to predict the behavior of such materials, and to ultimately put this knowledge to use in industry and technology, it is necessary to bridge this multitude of scales by appropriate schemes of articulated theoretical, numerical, and experimental approaches. This proposal is focused on the theoretical side of this venture, with the aim to contributing to the identification of problems of national scientific importance that offer new opportunities for the integration of applied analysis in research and in the education of advanced graduate students and postdoctoral fellows.