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.

偏微分方程 (PDE) 是用于模拟电磁学、天文学和流体动力学等自然现象的数学工具，该项目旨在了解此类方程的解如何表现，拉普拉斯方程是典型的椭圆 PDE。 ，用于对稳态齐次系统进行建模，该方程在偏微分方程、复分析、调和分析、几何和工程领域进行了研究，因此也研究了其解的行为（称为调和）；然而，关于更复杂的方程（例如模拟量子行为、具有微观结构的系统以及随时间变化的系统）的解的行为，仍然存在许多问题。用于回答这些问题，从而增进椭圆和抛物型偏微分方程领域的知识。出于增加代表性不足群体的参与以及解决学术界保留的常见问题的目标，该项目整合了包容性的内容。偏微分方程和调和分析研讨会的目标受众将包括处于职业生涯艰难过渡阶段的初级数学家，特别是那些来自历史上代表性不足的群体的演讲者将被选择以反映学生参与者的人口统计数据和更大多样性的潜力。拉普拉斯方程是一个在真正均匀的环境中模拟稳态现象的偏微分方程，由于我们生活的世界不是各向同性的真空，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，薛定谔方程描述了量子力学波的行为，而它的概括则描述了更复杂的设置，因此，需要了解一般椭圆偏微分方程解的性质。具体来说，研究人员将围绕使用调和函数的已知属性来更好地理解椭圆方程的解，探索变量系数和低阶项的存在如何影响椭圆方程的解的行为。鉴于像热方程这样的抛物线方程可以模拟随时间变化的系统的演化，因此了解此类偏微分方程的解如何表现也很重要。另一方面，研究人员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，研究人员将构建一个在高维环境中使用椭圆理论来理解抛物方程解的性质的框架。该奖项反映了。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。