CAREER: Elliptic and Parabolic Partial Differential Equations

偏微分方程（PDE）是用于模拟自然现象的数学工具，例如电磁学，天文学和流体动力学。这个项目关注的是理解这些方程的解是如何表现的。拉普拉斯方程是典型的椭圆型偏微分方程，它被用来模拟稳态齐次系统。这个方程在偏微分方程、复分析、调和分析、几何学和工程学领域都有研究;因此，它的解（称为调和函数）的行为是非常好理解的。然而，关于更复杂方程的解的行为仍然存在许多问题，例如那些模拟量子行为的方程，具有微观结构的系统以及随时间变化的系统。研究者的调和函数知识将用于回答这些问题，从而推进椭圆和抛物偏微分方程领域的知识。出于增加代表性不足的群体的参与，以及解决学术界保留的共同问题的目标，该项目整合了PDE和谐波分析的包容性研讨会。讲习班的目标受众将包括在职业生涯中处于困难过渡阶段的初级数学家，特别是那些历史上代表性不足的群体。演讲者的选择将反映学生参与者的人口统计数据和我们学科更大的多样性的潜力。拉普拉斯方程是一个PDE，在一个真正均匀的环境中模拟稳态现象。由于我们生活的世界不是一个各向同性的真空，支配许多自然现象的数学方程往往比拉普拉斯方程更复杂。例如，薛定谔方程描述了量子力学波的行为，而它的推广描述了更复杂的设置。因此，有必要了解一般椭圆偏微分方程的解决方案的性质。这个研究项目的一个组成部分围绕使用调和函数的已知属性，以获得更好的理解椭圆方程的解决方案。具体来说，调查员将探讨如何存在的变系数和低阶项影响椭圆方程的解决方案的行为。这条调查线将通过独特的延续和同质化理论的角度来解决。考虑到抛物型方程（如热方程）对随时间变化的系统的演化进行建模，理解此类PDE的解如何表现也很重要。因此，在另一个方向，研究者将使用椭圆理论来解决与抛物型偏微分方程相关的问题。更具体地说，研究者将构建一个框架，在高维环境中使用椭圆理论来理解抛物方程解的性质。该奖项反映了NSF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。

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.