Nonlocal Variational Problems from Physical and Biological Models

非局部模型用于描述材料科学、量子力学、数学物理和生物学中的各种物理现象。这些模型的相关性在于，它们引入了用于研究宏观领域微观结构的尺度，就像描述生物系统集体行为的非局部变分模型一样。该项目涉及非局部吸引-排斥相互作用能的数学分析，这些相互作用能与生物和机器人群体、颗粒介质和纳米粒子自组装的聚集模型直接相关。重点是理解相互作用对所考虑的物理系统的竞争效应。这将深入了解非定域性的一般现象学，并最终提供这些系统中集体行为的预测，并指导改进设备的设计。将本科生和研究生的培养整合到研究项目中，通过学生的研究项目发现新的成果。该项目有三个主要的数学目标：(1)开发新的工具来研究描述相反带电相分布的模型的优化器的对称性；(2)研究了用吸引-排斥非局域能量描述的群体模型中界面自由能对正则化的影响；最后，(3)引入伽莫夫液滴模型的扩展，以研究中子在决定原子核形状和核裂变阈值方面的作用。为了实现这些目标，将采用从非线性到几何分析、优化和偏微分方程的各种技术。这些技术将与基于计算机的实验相结合。这三个方向的共同目标是理解各种实际感兴趣的物理系统中的优化器结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Nonlocal models are used to describe a wide array of physical phenomena in material science, quantum mechanics, mathematical physics, and biology. The pertinence of these models is that they introduce scales used to investigate microstructures in macroscopic domains, as in the case of the nonlocal variational models characterizing the collective behavior in biological systems. This project involves the mathematical analysis of nonlocal attractive-repulsive interaction energies that are directly connected to aggregations models for biological and robotic swarming, granular media, and self-assembly of nanoparticles. The emphasis is on understanding the competing effects of interactions on the physical systems considered. This will yield insight into the general phenomenology of nonlocality and ultimately provide predictions of collective behavior in these systems and guide the design of improved devices. Training of undergraduate and graduate students will be integrated in the research project, which will lead to the discovery of new results through student research projects. The project has three main mathematical aims: (1) Develop new tools to study symmetry of optimizers of a model describing the distribution of oppositely charged phases; (2) study the effect of regularization via an interfacial free energy in swarming models described by attractive-repulsive nonlocal energies; and, finally, (3) introduce an extension of Gamow’s liquid drop model to investigate the effect of neutrons in determining the shape of the nucleus of an atom and the threshold of nuclear fission. To pursue these goals, techniques varying from nonlinear to geometric analysis, optimization and partial differential equations will be implemented. These techniques will be combined with computer-based experimentation. The common aim of all three directions is understanding structures of optimizers in a variety of physical systems of practical interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.