Nonlinear Partial Differential Equations and Geometry

非线性偏微分方程和几何

• 批准号: 2005311
• 负责人: Benjamin Weinkove
• 金额: $ 24.79万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005311&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Geometry; Nonlinear; Partial; Differential; Equations; Geometry

Many physical theories have been modeled successfully by mathematical equations known as partial differential equations (PDEs). The classical examples are the heat equation, the wave equation and the Laplace equation, which describe in ideal conditions the behavior of heat, waves and electrostatic potentials respectively. These are linear PDEs. More complicated theories are often modeled by nonlinear PDEs. A classic nonlinear PDE is the Monge-Ampere equation which has connections to the physical theory of optics but is also a powerful tool in the study of geometry. This research will investigate several nonlinear PDE, all related to the Monge-Ampere equation, which arise in geometry or which exhibit geometric behavior. The research goals are two-fold: to use nonlinear PDEs to advance our understanding of geometric spaces and the structures that live on them; and to understand the phenomena and behavior of solutions of nonlinear PDE using the tools and language of geometry. In addition the PI will also train graduate students in the methods of nonlinear PDEs and geometry, and guide their research.The PI will carry out research on nonlinear PDEs and geometry. There are five projects, linked by the common theme of the complex Monge-Ampere equation and focusing on developing new analytic tools and strategies to derive a priori estimates. For the first project, the PI will investigate a conjecture of Donaldson extending Yau’s theorem on the complex Monge-Ampere equation to the setting of symplectic 4-manifolds, using an ansatz which reduces the nonlinear PDE to an equation of a single real-valued function. A second project will study Perelman’s estimates for the Kahler-Ricci flow with the goal of understanding the behavior of the flow when there is a finite time singularity. A further project is to investigate a non-Kahler version of this flow, known as the Chern-Ricci flow, with a focus on finite time singularities on complex surfaces. Another project will extend the work of Chen-Cheng and Shen on the question of existence of metrics with constant Chern scalar curvature, in the setting of non-Kahler complex surfaces. A final project is to study the uniform convexity of convex solutions to PDEs satisfying certain structure conditions, building on constant rank theorems.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.

许多物理理论已通过称为偏微分方程（PDE）的数学方程式成功建模。经典的例子是热方程，波动方程和拉普拉斯方程，它们在理想条件下分别描述了热，波和静电电位的行为。这些是线性PDE。更复杂的理论通常由非线性PDE建模。经典的非线性PDE是Monge-Ampere方程，它与光学物理理论有联系，但也是研究几何学研究的强大工具。这项研究将研究几种与Monge-Ampere方程相关的非线性PDE，这些方程在几何形状或暴露的几何行为中产生。研究目标是两个方面：使用非线性PDE来促进我们对几何空间及其生活结构的理解；并使用几何形状的工具和语言了解非线性PDE解决方案的现象和行为。此外，PI还将培训研究生的非线性PDE和几何学方法，并指导他们的研究。PI将对非线性PDES和几何形状进行研究。有五个项目，由复杂的Monge-Ampere方程的共同主题链接，并着重于开发新的分析工具和策略来得出先验估计。对于第一个项目，PI将使用ANSATZ将Yau关于复杂的Monge-Ampere方程的Yau定理扩展到符号4 manifolds的设置的概念，该定理使用ANSATZ将非线性PDE降低到单个实数函数的方程式。第二个项目将研究Perelman对Kahler-Ricci流量的估计，目的是在有限的时间奇点时了解流动的行为。另一个项目是研究这种流的非卡勒版本，即Chern-Ricci流，重点是在复杂表面上的最终时间奇点。另一个项目将在非Kahler复合物表面的环境中扩展Chen-Cheng and Shen关于具有持续的Chern标量曲率的指标的问题。最终项目是研究凸溶液的均匀凸，以满足某些结构条件的PDE，并以恒定等级定理为基础。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估值得支持。