CAREER: Integration of Research and Education in the Study of Analysis and Partial Differential Equations in Carnot-Caratheodory Spaces

摘要--------本提案的研究部分涉及与Carnot-Caratheodory空间几何有关的某些非线性偏微分方程组的解的研究。更准确地说，我们将研究拟线性亚椭圆系统解的锐正则性，并将结果应用于卡诺群和更一般空间中的拟正则映射理论。我们将研究海森堡群中有目标映射和黎曼流形或闵可夫斯基空间中有区域映射的能量临界点。分析了亚椭圆算子的边值问题，研究了非发散形式非线性方程解的定性和定量性质。该提案的教育部分以创建综合研究小组为中心，在该小组中，本科生和研究生在PI的监督下进行基础研究项目。这项活动的目标之一是通过网络数据库、讨论会和出版物传播信息。偏微分方程用于为现实生活中的现象提供数学模型，如化学反应、力场或电导率。这样的模型既可以更好地理解现象，也可以尝试预测所研究系统的未来行为。粗略地说，PDE的光滑解对应于可以很好地理解和建模的系统。对解的正则性的研究在于寻找光滑解。在这个提议中描述的方程出现在研究约束下的运动（如机器人手臂运动），以及研究非各向同性介质中的扩散。

ABSTRACT--------The research component of this proposal deals with the study of solutions to certain non-linear systems of Partial DifferentialEquations which arise in connections with the geometry of Carnot-Caratheodory spaces. More precisely, we will study the sharp regularity of solutions to quasilinear subelliptic systems, and apply the results to the theory of quasi-regular maps in Carnot groups and in more general spaces. We will study critical points of the energy for maps with target in the Heisenberg group and with domain in a Riemannian manifold or in the Minkowski space.The analysis of boundary value problems for sub-elliptic operators, and the study of qualitative and quantitative properties of solutions of some non-divergence form non-linear equations, are also part of this proposal.The educational component of this proposal is centered around the creation of Integrated Research Groups, in which undergraduate and graduate students work on basic research projects supervised by the PI. One of thegoals of this activity is the dissemination of information, through a web-based data-base, seminars, and publications. Partial differential equations are used to provide a mathematical model for real-life phenomena, such as chemical reactions, force-fields or conductivity. Such modelsallow both to better understand the phenomena and to try to forecast the future behavior of the system under study. Roughly speaking, smooth solutions to PDE's correspond to systems which can be well understood and modeled. The study of regularity of solutions consists in finding smooth solutions.The equations described in this proposal appear in the study ofmotion under constraints (as in Robot arms motion),and in the study of diffusion in a non isotropic media.