Geometric nonlinear partial differential equations

几何非线性偏微分方程

• 批准号: 46732-2010
• 负责人: Guan, Pengfei
• 金额: $ 2.91万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635317
• 关键词: Geometric; nonlinear; partial; differential; equations

Differential geometry and differential equations are involved in many practical problems and often highly theoretical problems in Physics and Engineering and other areas. Most of these problems are guided by nonlinear differential equations. For example, the string theory in theoretical physics, where differential geometry and differential equations play important roles. Various ideas of analysis are used in the process of finding solutions of differential equations with certain geometric properties in general and in particular those that are nonlinear. The main thrusts in differential geometry are the search for ``optimal" geometric structures, such as diffeomorphisms, metrics, etc., and the study of the geometric and topological implications of their existence. These ``extremal" geometric objects can be viewed as solutions to natural nonlinear partial differential equations, and they encode rich information linking geometry, topology and analysis. Curvature tensors yield important examples, e.g. Ricci tensor and Weingarten map. The study of these curvature tensors in general carried out through systems of parabolic and elliptic nonlinear equations. The examples including the Monge-Amp\`ere equations, Gauss curvature flow, and the Ricci flow. We continue to study the fully nonlinear partial differential equations related to problems in differential geometry.A common thread linking our program is the analysis of the geometric fully nonlinear equations and their relationship with geometric quantities . The fundamental existence and regularity questions in the category of analysis should be adapted to cope with the emphasis on {\it geometric} solutions, which are often forced upon us by the geometric nature of the problems like the monotonicity of specified geometric functionals. Our objective is to develop various analytic tools to establish a priori estimates for these equations, explore the structures of geometric solutions, and derive geometric consequences.

微分几何和微分方程涉及到物理和工程等领域的许多实际问题，而且往往是理论性很强的问题。这些问题大多是由非线性微分方程引导的。例如，理论物理学中的弦理论，其中微分几何和微分方程扮演着重要角色。各种各样的分析思想被用于寻找具有某些几何性质的微分方程的解的过程中，特别是那些非线性的。微分几何的主要目标是寻找"最优”几何结构，如同构、度量等，以及研究它们存在的几何和拓扑含义。这些“极值”几何对象可以被看作是自然非线性偏微分方程的解，它们编码了丰富的信息，将几何、拓扑和分析联系起来。曲率张量给出了重要的例子，例如Ricci张量和Weingarten映射。这些曲率张量的研究一般通过抛物和椭圆非线性方程组进行。例子包括Monge-Schwarzere方程、Gauss曲率流和Ricci流。我们继续研究与微分几何问题相关的完全非线性偏微分方程，一个共同的主线是分析几何完全非线性方程及其与几何量的关系。分析范畴中的基本存在性和正则性问题应该适应于强调几何解，这通常是由问题的几何性质（如指定几何泛函的单调性）强加给我们的。我们的目标是开发各种分析工具来建立这些方程的先验估计，探索几何解的结构，并得出几何后果。