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

研究者开发了新的方法，需要研究关键问题的新兴领域的变分法在L-无穷大的情况下，泛函被最小化的一些功能的字段是受常秩微分约束的基本上确界。 这个一般框架包括治疗的卷曲自由领域，发散自由领域，对称化和高阶梯度，并满足麦克斯韦方程组的领域。 他研究的必要和充分条件下，这些功能具有极小，放松，最高代表性，均匀化，近似通过伽玛收敛，他调查选定的系统的偏微分方程出现的阿龙松方程相关的这些功能。 以前在这个方向上的工作一直集中在特定的情况下，可容许的领域是梯度（无旋）;相应的问题上的上泛函的领域受到其他微分约束，自然出现的平衡法仍然在很大程度上没有解决。 该项目的一个重要目标是弥合这一差距，追求系统的研究，在这种普遍性的超泛函。 该项目的目的是提供一个统一的观点的主题，特别是，揭示如何已知的性质取决于梯度的上泛函可以被理解为一个特定的情况下，在更广泛的背景下考虑。 由研究者和他的合作者开发的分析方法用于研究材料科学中的选定问题，这些问题通过涉及至上泛函的显式变分原理来描述。 特别是，该项目旨在回答有关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，其中新模型被分析并用于研究在相关Monge-Kantorovich传质问题的背景下，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材料的行为的具体问题。 这些材料非常常见：自然界中发现的大多数金属都是多晶体;谷物，煤炭，塑料，建筑材料（沙子，砾石）以及各种粉末和化合物都是日常处理和储存的颗粒材料的例子。 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，如非线性弹性、图像处理、损伤和断裂力学、塑性和半导体设计中，普遍存在具有实际重要性的问题，其中更现实的方法是最小化上确界而不是某些量的平均值。 预计在本项目中开发的方法将与广泛的应用有关，超出本文明确考虑的范围。 还通过培训研究生和本科生，使他们了解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，实现项目的更广泛影响。

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.