Nonlinear Subelliptic Analysis

非线性亚椭圆分析

• 批准号: 0500983
• 负责人: Juan Manfredi
• 金额: --
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500983&HistoricalAwards=false
• 关键词: Nonlinear; Subelliptic; Analysis

ABSTRACT A major part of the success in the linear and quasi-linear theory comes from interpreting derivatives in the generalized sense of distributions, allowing for a more powerful calculus. Distributions do not seem, in general well suited to non-linear problems because they cannot be multiplied. Jets are generalized (local) point-wise derivatives that allow for the interpretation and calculation of non-linear functions of derivatives. In this proposal, the PI presents a project touse jets to develop some basic Analysis tools in general state spaces. Typically, in these spaces higher derivatives with respect different parameters do not commute, as in the Euclidean case, but rather satisfy more complicated algebraic relations. Jets adapted to the geometry of a state space endowed with a family of vector fields satisfying a non-degeneracy condition are called sub-elliptic jets. Basic analysis topics like Taylor developments and maximum principles have to be adapted to conform to the new sub-elliptic geometry. Topics studied include: sub-elliptic extensions of the uniqueness theorem of R. Jensen for viscosity solutions, regularity for the sub-elliptic p-Laplacian, sub-elliptic convex functions, and Cordes sub-elliptic estimates. The derivative is a basic tool in mathematical analysis, used to measure the growth and decay of functions. Knowledge of the derivative of a function allows for its recovery by means of integration. When trying to model complex scientific phenomena it is often necessary to write down equations satisfied by derivatives, and derivatives of derivatives, of functions with respect to several parameters. These equations are called partial differential equations. In this proposal the PI proposes to develop tools to study partial differential equations written in terms of vector fields. These equations have applications to problems in Robotics, Control Theory and Mathematical Finance.

摘要线性和准线性理论的成功的主要部分来自在广义分布意义上解释衍生物，从而实现了更强大的演算。通常，分布似乎不适合非线性问题，因为它们不能乘以它们。喷气机是普遍的（局部）点衍生物，可以解释和计算衍生物的非线性函数。在此提案中，PI提出了一个项目的喷气机，以在一般状态空间中开发一些基本分析工具。通常，在这些空间中，具有不同参数的较高衍生物不通勤，例如在欧几里得的情况下，而是满足更复杂的代数关系。适合于满足非平稳条件的矢量场的状态空间的几何形状的喷气机称为亚椭圆形射流。基本分析主题（例如泰勒发展和最大原则）必须适应符合新的亚椭圆形几何形状。研究的主题包括：用于粘度溶液的R. Jensen唯一性定理的亚纤维化扩展，纤维化p-laplacian的规律性，亚纤维化凸函数和电线亚纤维化估计值。衍生物是数学分析的基本工具，用于衡量功能的生长和衰减。了解功能的导数的知识可以通过集成恢复。在试图建模复杂的科学现象时，通常有必要写下有关几个参数的衍生物和衍生物衍生物满足的方程。这些方程称为部分微分方程。在此提案中，PI建议开发工具来研究根据向量领域编写的部分微分方程。 这些方程在机器人技术，控制理论和数学金融中有应用。