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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=593182
• 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.

差异几何形状和微分方程参与了许多实际问题，并且在物理和工程以及其他领域中通常是高度理论问题。这些问题大多数都以非线性微分方程为指导。例如，理论物理学中的弦理论，其中差分几何和微分方程起着重要作用。在寻找具有某些几何特性的微分方程解决方案的解决方案的过程中，尤其是非线性的分析思想。差异几何形状中的主要推力是寻找``最佳''几何结构，例如差异性，指标等，以及对其存在的几何和拓扑含义的研究。这些``极端''几何对象可被视为自然非线性偏微分方程的解决方案，它们编码了链接几何形状，拓扑和分析的丰富信息。曲率张量产生重要的例子，例如Ricci Tensor和Weingarten地图。一般来说，对这些曲率张量的研究通过抛物线和椭圆形非线性方程式进行。包括Monge-Amp \'eRE方程，高斯曲率流和Ricci流中的示例。我们继续研究与差异几何学问题有关的完全非线性偏微分方程。