In recent years there has been remarkable progress in the mathematical rigorous understanding of nonlinear phenomena, resulting from a deep articulation of ideas and techniques from partial differential equations, calculus of variations, and geometric measure theory. Common features to the problems to be addressed in this project are the treatment of energies that involve terms of different dimensionality (bulk, surface terms, etc.), a large range of length and time scales, higher order derivatives, and discontinuous underlying fields. Topics include: - in imaging, the recolorization and reconstruction of damaged images using the RGB and the CB models; - in thin structures, the understanding of the interplay between rigidity and brittle features; - within the realm of singular perturbations, the study of foams, microphase separation of copolymer melts, relaxation and homogenization of multi-phase, multi-component multiple integrals, micromagnetics; - in epitaxially strained thin films, the surface morphology and diffusion; - in micromagnetics, the derivation of a model for large bodies from the small bodies model that exhibits competition between the anisotropic energy and the exchange energy terms; - in the calculus of variations, the development of multi-scale theories and dimension reduction techniques for systems that go beyond traditional (higher order) underlying gradient fields and may apply to Maxwell-type systems.
The program outlined above is strongly motivated by contemporary issues in imaging and materials science at the core of advances in high-end technology. These include recolorization of damaged images, the understanding of fracture in thin structures, the study of foams used in oil recovery, detergents, and lightweight structural materials, the study of morphology and defects in the epitaxial deposition process that are responsible for important optical, electronic, and magnetic properties, and the prediction of the behavior of ferroelectric, electromagnetic, and magnetostrictive materials and composites. The underlying models are at the forefront of traditional mathematical theories, and require state-of-the-art techniques, new ideas, and the introduction of innovative mathematical tools. It is necessary to bridge the multitude of scales present, and other mathematically challenging features, by appropriate schemes of articulated theoretical, numerical, and experimental approaches. This project is focused on the theoretical side of this venture, with the aim of 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.
