CAREER: Elliptic and Parabolic Partial Differential Equations

职业：椭圆和抛物型偏微分方程

• 批准号: 2236491
• 负责人: Blair Davey
• 金额: $ 49.87万
• 依托单位: Montana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2236491&HistoricalAwards=false
• 关键词: CAREER; Elliptic; Parabolic; Partial; Differential

Partial differential equations (PDE) are mathematical tools that are used to model natural phenomena like electromagnetism, astronomy, and fluid dynamics, for example. This project is concerned with understanding how the solutions to such equations behave. The Laplace equation is the prototypical elliptic PDE, and it is used to model steady-state homogeneous systems. This equation is studied in the fields of PDE, complex analysis, harmonic analysis, geometry, and engineering; and therefore, the behavior of its solutions (known as harmonic functions) is very well-understood. However, many questions remain regarding the behavior of solutions to more complicated equations like those that model quantum behavior, systems with microscopic structure, and systems that are changing in time. The investigator’s knowledge of harmonic functions will be used to answer these questions, thereby advancing knowledge in the areas of elliptic and parabolic partial differential equations. Motivated by the goal of increasing participation from underrepresented groups, as well as addressing common issues with retention in academia, this project integrates an inclusive workshop in PDE and harmonic analysis. The target workshop audience will include junior mathematicians who are at difficult transitional stages in their careers, especially those from historically underrepresented groups. Speakers will be chosen to reflect the demographics of the student participants and the potential for greater diversity in our discipline.The Laplace equation is a PDE that models steady-state phenomena in a truly uniform environment. Since the world that we live in is not an isotropic vacuum, the mathematical equations that govern many natural phenomena are often more complicated than Laplace’s equation. For example, the Schrodinger equation describes the behavior of quantum-mechanical waves, while its generalizations describe even more complex settings. As such, there is a need to understand the properties of solutions to general elliptic PDEs. One component of this research project revolves around using known properties of harmonic functions to gain a better understanding of solutions to elliptic equations. Specifically, the investigator will explore how the presence of variable coefficients and lower-order terms affects the behavior of solutions to elliptic equations. This line of inquiry will be addressed through the perspectives of unique continuation and homogenization theory. Given that parabolic equations like the heat equation model the evolution of systems that are changing in time, it is also important to understand how the solutions to such PDE behave. Therefore, in another direction, the investigator will use elliptic theory to tackle problems related to parabolic PDE. More specifically, the investigator will construct a framework for using elliptic theory in high-dimensional settings to understand the properties of solutions to parabolic equations.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.

部分微分方程（PDE）是数学工具，用于模拟自然现象，例如电磁，天文学和流体动力学。该项目关注的是理解该方程式的解决方案。拉普拉斯方程是原型椭圆PDE，用于模拟稳态均匀系统。该方程在PDE，复杂分析，谐波分析，几何和工程领域进行了研究。因此，其解决方案的行为（称为谐波功能）是非常理解的。但是，关于解决方案对更复杂方程的行为的行为，例如建模量子行为，具有微观结构的系统以及随时间变化的系统的行为。研究者对谐波功能的知识将用于回答这些问题，从而在椭圆形和抛物线偏微分方程方面的知识中提高知识。由于代表性不足的群体的参与以及解决学术界保留的常见问题的目的，该项目的推动力集成了PDE和Harmonic分析中的包容性研讨会。目标讲习班的观众将包括年轻的数学家，他们的职业生涯中处于艰难的过渡阶段，尤其是从历史上代表性不足的群体中。将选择演讲者来反映学生参与者的人口统计信息，并在我们的学科中获得更大的多样性。拉普拉斯方程是在真正统一的环境中建模稳态现象的PDE。由于我们生活的世界不是各向同性真空，因此控制许多自然现象的数学方程通常比拉普拉斯方程更复杂。例如，Schrodinger方程描述了量子力波的行为，而其概括描述了更复杂的设置。因此，有必要了解通用椭圆PDE的解决方案的特性。该研究项目的一个组成部分围绕着使用谐波功能的已知特性，以更好地了解椭圆方程的解决方案。具体而言，研究者将探讨可变系数和低阶项的存在如何影响椭圆方程的解决方案的行为。这种探究线将通过独特的延续和同质化理论的角度来解决。鉴于诸如热方程模型之类的抛物线方程在时间变化的系统的演变中，也重要的是要了解该PDE的解决方案如何行为。因此，在另一个方向上，研究者将使用椭圆理论来解决与抛物线PDE有关的问题。更具体地说，研究人员将构建一个在高维环境中使用椭圆理论的框架，以了解抛物线方程解决方案的特性。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准通过评估来获得的支持。