Modern Methods in Calculus of Variations with Applications to Polycrystalline and Granular Materials

研究人员发展了一种新的方法来研究L无穷维变分学新兴领域中的关键问题，即当所要最小化的泛函是受常数阶微分约束的场的某些函数的本质上确界时。这一一般框架包括处理无旋度场、无散度场、对称化和高阶梯度，以及满足麦克斯韦方程的场。他研究了这些泛函具有极小化、松弛、最高表示、齐次化、伽玛收敛逼近的充要条件，并研究了作为与这些泛函相关的Aronsson方程的偏微分方程组。以前在这方面的工作主要集中在允许的场是梯度(无旋度)的特殊情况；作用于受其他微分约束的场的最高泛函的相应问题基本上没有得到解决。该项目的一个重要目标是通过系统地研究这一一般性中的最高泛函来弥合这一差距。这个项目的目的是提供一个主题的统一视图，特别是揭示如何在更广泛的背景下将依赖于梯度的最高泛函的已知性质理解为特定的情况。研究人员和他的合作者发展的分析方法被用来研究材料科学中选定的问题，这些问题是通过涉及最高泛函的显式变分原理来描述的。特别是，该项目试图回答关于多晶体宏观行为的某些问题，并阐明在颗粒材料研究中出现的几个问题，其中新的模型被分析并用于在相关的Monge-Kantorovich质量转移问题的背景下研究在给定景观上的异质沙堆的流动。该项目的动机是关于多晶和颗粒状材料行为的具体问题。这类材料非常常见：自然界中发现的大多数金属都是多晶体；谷物、煤、塑料、建筑材料(沙、砾石)以及各种粉末和化合物都是日常处理和储存的颗粒状材料。因此，了解如何以最佳方式混合和运输它们是一个自然感兴趣的问题。在许多其他领域，例如非线性弹性、图像处理、损伤和断裂力学、塑性和半导体设计，更现实的方法是最小化上界而不是某些量的平均值的实际重要问题是普遍存在的。预计在本项目中开发的方法将适用于广泛的应用，而不是这里明确考虑的那些。该项目的更广泛影响也是通过培训研究生和本科生实现的，这些学生接触到应用分析的现代发展，并参与与提案有关的项目，以及通过其他教育和外联活动。

The investigator develops new methods needed to study key issues in the emerging area of Calculus of Variations in L-infinity, for the case where the functionals to be minimized are the essential supremum of some function of fields that are subject to constant rank differential constraints. This general framework includes the treatment of curl-free fields, divergence-free fields, symmetrized and higher order gradients, and fields that satisfy Maxwell's equations. He studies necessary and sufficient conditions under which these functionals possess a minimizer, relaxation, supremal representation, homogenization, approximation via Gamma-convergence, and he investigates selected systems of partial differential equations that arise as the Aronsson equations associated to such functionals. Previous work in this direction has been focused on the particular case where the admissible fields are gradients (curl-free); the corresponding issues for supremal functionals acting on fields subject to other differential constraints that appear naturally as balance laws have remained largely unaddressed. An important goal of the project is to bridge this gap by pursuing a systematic study of supremal functionals in this generality. The project aims to provide a unified view of the subject and, in particular, to reveal how the known properties of supremal functionals depending on gradients can be understood as a particular case in the broader context considered. The analytical methods developed by the investigator and his collaborators are used to study selected problems in materials sciences that are described by means of explicit variational principles involving supremal functionals. In particular, the project seeks to answer certain questions regarding the macroscopic behavior of polycrystals, and to elucidate several issues that come up in the study of granular materials, where new models are analyzed and used to study the flow of heterogeneous sand piles over given landscapes in the context of related Monge-Kantorovich mass transfer problems. The project is motivated by specific questions about the behavior of polycrystalline and granular materials. Such materials are very common: most metals found in nature are polycrystals; grains, coal, plastic, building materials (sand, gravel), and various powders and chemical compounds, are all examples of granular materials that are handled and stored on a daily basis. Thus, understanding how to mix and transport them in an optimal fashion is an issue of natural interest. Problems of practical importance where the more realistic approach is to minimize the supremum rather than the average of certain quantities are ubiquitous in many other areas, such as nonlinear elasticity, image processing, damage and fracture mechanics, plasticity, and semiconductor design. It is expected that the methods developed within this project will be relevant to an extensive range of applications, beyond those considered here explicitly. The broader impacts of the project are also achieved through training of graduate and undergraduate students, who are exposed to modern developments in applied analysis and are involved in projects related to the proposal, as well as through other educational and outreach activities.