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Collaborative Research: The Structure of Nonlocal Operators and Applications

合作研究：非本地算子的结构和应用

基本信息

批准号：
1665285
负责人：
Russell Schwab
金额：
$ 15万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-05-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665285&HistoricalAwards=false
关键词：
Collaborative Research Structure Nonlocal Operators

项目摘要

This project seeks to answer questions about, to develop an improved mathematical understanding of, and to build more sophisticated tools involved in the modeling of phenomena driven by nonlocal interactions. A nonlocal interaction can be illustrated by its contrast to a local interaction in a simplified picture of a population of agents (say, people) interacting across a shared network. A local model is one in which people can share information only with their immediate neighbors, whereas a nonlocal model is one in which they can share information more broadly, possibly across an entire network, and do so instantly. This simplified picture actually manifests itself in many situations in which a nonlocal paradigm is more fruitful than a local one, and the inclusion of nonlocal interactions often fundamentally changes the underlying mathematics that describe aspects such as the propagation of information. This project aims to improve the study of such phenomena through the introduction of new mathematical tools. Along the way, this project will support the mentoring of graduate students and undergraduates through involvement in research on these topics. It will also create new graduate course material that provides a more unified approach to nonlocal tools of use in treating certain current topics in data science. In the realm of mathematics surrounding nonlocal problems, there has been a surge of activity during the past twenty years. Experts have found themselves addressing questions in roughly two categories: (1) What properties of nonlocal equations should be studied for the sake of nonlocality itself; and (2) what properties of nonlocal equations should be studied for the sake of integrating them with other fields? In this project the principal investigators focus on the second type of question. They aim to develop nonlocal tools that can be applied to some classical problems that are not, at first glance, necessarily nonlocal, including questions about oscillatory boundary-layer phenomena for elliptic equations and free-boundary problems of one or two phases. The key point is that there are powerful tools in the nonlocal world that could shed new light upon or produce new results for some of these equations that were not approached from this perspective earlier. Such tools include, but are not limited to, the fast growing regularity theory for nonlocal equations and theory of weak solutions for nonlocal equations (still in its infancy). Through representation techniques for general (nonlinear) operators that enjoy a global comparison principle, the principal investigators hope to bridge the gap between some aspects of these previously disjoint classes of equations and to use this "nonlocal" bridge as a pathway to new discoveries.

这个项目寻求回答关于非局部相互作用所驱动的现象的建模的问题，发展对非局部相互作用的更好的数学理解，并建立更复杂的工具。非本地交互可以通过它与本地交互的对比来说明，在简化的图片中，一群代理(比方说，人)通过共享网络交互。本地模式是人们只能与他们的近邻共享信息的模式，而非本地模式是他们可以更广泛地共享信息的模式，可能是跨整个网络，并立即这样做。这种简化的图景实际上体现在许多情况下，其中非本地范例比本地范例更有成效，并且包括非本地交互通常从根本上改变了描述诸如信息传播等方面的基本数学。该项目旨在通过引入新的数学工具来改进对这种现象的研究。在此过程中，该项目将通过参与这些主题的研究来支持对研究生和本科生的指导。它还将创建新的研究生课程材料，为在处理数据科学的某些当前主题时使用非本地工具提供更统一的方法。在围绕非局部问题的数学领域，在过去的二十年里，活动激增。专家们发现他们研究的问题大致分为两类：(1)为了非局域性本身，应该研究非局部方程的什么性质；(2)为了将非局部方程与其他领域结合起来，应该研究非局部方程的什么性质？在这个项目中，主要调查人员侧重于第二类问题。他们的目标是开发非局部工具，这些工具可以应用于一些乍看起来不一定是非局部的经典问题，包括关于椭圆型方程的振荡边界层现象的问题，以及单相或两相的自由边界问题。关键的一点是，在非本地世界中，有一些强大的工具可以为这些方程中的一些先前没有从这个角度进行探讨的问题提供新的线索或产生新的结果。这些工具包括但不限于，非局部方程的快速增长的正则性理论和非局部方程的弱解理论(仍处于起步阶段)。通过具有全局比较原理的一般(非线性)算子的表示技术，主要研究人员希望弥合这些以前互不相交的方程类的某些方面之间的差距，并将这种“非局部”桥梁用作通向新发现的途径。