Relaxation and Regularity Theory in the Calculus of Variations: Applications to Multiscale Problems, Thin Structures, and Magnetic Materials

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DMS Award AbstractAward #: 0103799PI: Fonseca, IreneInstitution: Carnegie Mellon University Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Relaxation and Regularity Theory in the Calculus of Variations: Applications to Multiscale Problems, Thin Structures, and Magnetic MaterialsThe general objective of this project is the development of techniques in the calculus of variations, geometric measure theory, and in the regularity theory for systems of partial differential equations to address equilibrium and stability problems involving both bulk and surface energies, and where the admissible fields develop fast, multiple scale oscillations, and well as defect concentrations. The underlying models include singularly perturbed energies, multiscale homogenization, shape optimization for gradient-constrained functionals, and multiscale problems for dimension reduction.The research activity will be motivated by contemporary issues in materials science and solid physics, where the contribution of mathematicians has already paved the way to important advances in the theoretical understanding of advanced materials and in high-technology performance. Emerging issues require state-of-the-art techniques in applied analysis, new ideas, and the introduction of innovative tools. The program contemplates the study of phase transformations (e.g. nucleation and growth of phases, dynamics of phase boundaries, multi-phase elasto-plastic materials), micromagnetism and ferromagnetism, nanostructures and thin films, optimal design of composites, and multiple scale problems.Date: May 30, 2001