Collaborative Research: The Structure of Nonlocal Operators and Applications

合作研究：非本地算子的结构和应用

• 批准号: 1700307
• 负责人: Nestor Guillen
• 金额: $ 13.5万
• 依托单位: University of Massachusetts Amherst
• 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 2020-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700307&HistoricalAwards=false
• 关键词: Collaborative; Research; Structure; Nonlocal; Operators

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.

该项目旨在回答有关的问题，对改进的数学理解，并建立更复杂的工具，其中涉及非本地互动驱动的现象的建模。 非局部互动可以通过与局部互动形成鲜明对比的相互作用来说明，在简化的图片中，跨共享网络相互作用的代理人（例如，人们）。 当地模型是人们只能与直接邻居共享信息的模型，而非本地模型是他们可以在整个网络中更广泛地共享信息并立即进行此信息。 这种简化的图片实际上在许多情况下表现出了自己的表现，在许多情况下，非局部范式比本地范式更富有成果，而且包含非局部相互作用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善对这种现象的研究。 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，为非本地使用工具提供更统一的方法，以处理数据科学中某些当前主题。 在围绕非局部问题的数学领域，在过去的二十年中，活动有一系列的活动。 专家发现自己在大约两个类别中解决了问题：（1）为了非局部性本身，应研究非局部方程的属性； （2）为了将其与其他字段集成在一起，应研究非局部方程的哪些属性？ 在这个项目中，主要研究人员专注于第二类问题。 他们的目的是开发可应用的非局部工具，这些工具乍一看不一定是非本地的经典问题，包括有关椭圆方程的振荡边界层现象的问题以及一个或两个阶段的自由边缘问题。关键点是，在非本地世界中有强大的工具可以为某些方程式提供新的启示或为这些方程式产生新的结果，而这些方程式从早些时候就没有接触过。这样的工具包括但不限于非本地方程的快速增长的规律性理论和非局部方程弱解决方案理论（仍处于起步期）。 通过对享有全球比较原则的一般（非线性）操作员的代表技术，主要研究人员希望弥合这些以前不一致的方程式类别的某些方面之间的差距，并使用这座“非局部”桥梁作为新发现的途径。

项目成果

Coupling Lévy measures and comparison principles for viscosity solutions

粘度解决方案的耦合 Lévy 测量和比较原理

• DOI: 10.1090/tran/7877
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Guillen, Nestor;Mou, Chenchen;Świȩch, Andrzej
• 通讯作者: Świȩch, Andrzej

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

• DOI: 10.1016/j.na.2019.02.021
• 发表时间: 2018-12
• 期刊: Nonlinear Analysis
• 作者: Nestor Guillen;Russell W. Schwab

Some free boundary problems recast as nonlocal parabolic equations

一些自由边界问题被改写为非局部抛物线方程

• DOI: 10.1016/j.na.2019.05.019
• 发表时间: 2019
• 期刊: Nonlinear Analysis
• 作者: Chang-Lara, Héctor A.;Guillen, Nestor;Schwab, Russell W.
• 通讯作者: Schwab, Russell W.

Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators

作为积分微分算子的狄利克雷到诺依曼映射的估计

• DOI: 10.1007/s11118-019-09776-w
• 发表时间: 2019
• 期刊: Potential analysis
• 影响因子: 1.1
• 作者: Guillen, Nestor;Kitagawa, Jun;Schwab, Russell W.
• 通讯作者: Schwab, Russell W.

Min–max formulas for nonlocal elliptic operators

• DOI: 10.1007/s00526-019-1631-z
• 发表时间: 2016-06
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Nestor Guillen;Russell W. Schwab