Nonlinear Subelliptic Analysis

线性和拟线性理论的成功很大程度上来自于对广义分布意义上的导数的解释，它允许更强大的微积分。一般来说，分布似乎不太适合非线性问题，因为它们不能相乘。射流是广义的（局部）逐点导数，允许解释和计算非线性函数的导数。在这个建议中，PI提出了一个使用喷气机在一般状态空间中开发一些基本分析工具的项目。通常，在这些空间中，对不同参数的高导数不会像在欧几里得情况下那样交换，而是满足更复杂的代数关系。具有满足非简并条件的一组向量场的状态空间的几何形状所适应的射流称为亚椭圆射流。基本的分析主题，如泰勒发展和极大原理必须适应新的亚椭圆几何。研究的主题包括：黏性解的R. Jensen唯一性定理的亚椭圆扩展，亚椭圆p- laplace的正则性，亚椭圆凸函数，以及Cordes亚椭圆估计。导数是数学分析中的一个基本工具，用于测量函数的增长和衰减。知道一个函数的导数，就可以用积分的方法恢复它。当试图为复杂的科学现象建模时，常常需要写出由函数的导数和导数的导数所满足的方程。这些方程被称为偏微分方程。在这个建议中，PI建议开发工具来研究用向量场表示的偏微分方程。这些方程适用于机器人、控制理论和数学金融等领域的问题。

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.