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

变分微积分中的松弛和正则理论：在多尺度问题、薄结构和磁性材料中的应用

• 批准号: 0103799
• 负责人: Irene Fonseca
• 金额: $ 23.37万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103799&HistoricalAwards=false
• 关键词: Relaxation; Regularity; Theory; Calculus; Variations

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

DMS Award AbstractAward #： 0103799 PI： Fonseca，Irene 卡内基梅隆大学 应用数学项目经理：Catherine Mavriplis职务：变分法中的松弛和正则性理论：应用于多尺度问题，薄结构和磁性材料这个项目的总目标是发展变分法，几何测量理论和偏微分方程系统的正则性理论，以解决涉及体积和表面能的平衡和稳定性问题，并且其中容许场快速发展、多尺度振荡以及缺陷浓度。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，以及降维的多尺度问题。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的理论理解和高技术性能的重要进展铺平了道路。新出现的问题需要应用分析、新思想和采用创新工具方面的最新技术。该计划设想相变的研究（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。