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主要来自于广义分布意义上的导数的解释，从而允许更强大的微积分。总的来说，发行版似乎并不适合 非线性问题，因为 它们不能被倍增。射流是广义（局部）逐点导数，允许解释和计算导数的非线性函数。在这个建议中，PI提出了一个项目，使用喷气机开发一些基本的分析工具，在一般的状态空间。通常，在这些空间中，关于不同参数的高阶导数不会像欧几里德那样交换，而是满足更复杂的代数关系。适应于具有满足非简并条件的一族矢量场的状态空间的几何形状的射流被称为亚椭圆射流。基本的分析主题，如泰勒发展和最大值原理，必须适应新的亚椭圆几何。研究的主题包括：R的唯一性定理的次椭圆扩展。詹森的粘性解决方案，正则性的亚椭圆p-Laplacian，亚椭圆凸函数，和Cordes亚椭圆估计。导数是数学分析中的一个基本工具，用来度量函数的增长和衰减。函数导数的知识允许通过积分来恢复函数。当试图模拟复杂的科学现象时，通常需要写下函数关于几个参数的导数和导数的导数所满足的方程。这些方程称为偏微分方程。在这个建议中，PI提出开发工具来研究向量场中的偏微分方程。 这些方程在机器人学、控制理论和数学金融中有应用。