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

奖项编号：0103799PI: Fonseca， IreneInstitution: Carnegie Mellon University项目：应用数学项目经理：Catherine mavriplis题目：变分学中的松弛与正则性理论在多尺度问题、薄结构和磁性材料中的应用本项目的总体目标是发展变分法、几何测量理论和偏微分方程系统的规则理论等技术，以解决涉及体能和表面能的平衡和稳定性问题，以及可容许场发展迅速、多尺度振荡和缺陷集中的问题。基础模型包括奇异摄动能量、多尺度均匀化、梯度约束泛函的形状优化和降维的多尺度问题。研究活动将受到材料科学和固体物理学的当代问题的推动，数学家的贡献已经为先进材料和高技术性能的理论理解方面的重要进展铺平了道路。新出现的问题需要最先进的应用分析技术、新思想和创新工具的引入。该计划考虑相变的研究（例如相的成核和生长，相边界的动力学，多相弹塑性材料），微磁性和铁磁性，纳米结构和薄膜，复合材料的优化设计，以及多尺度问题。日期：2001年5月30日

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