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

摘要-更确切地说，我们将研究拟线性次椭圆方程组解的尖锐正则性，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。我们将研究以Heisenberg群为目标，以Riemann流形或Minkowski空间为区域的映射的能量临界点，亚椭圆算子边值问题的分析，非发散型非线性方程解的定性和定量性质的研究，该提案的教育部分围绕着建立综合研究小组，在该小组中，本科生和研究生在PI的监督下从事基础研究项目。这项活动的目标之一是通过网上数据库、研讨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，例如化学反应，力场或导电性。这样的模型既可以更好地理解现象，又可以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。研究解的规律性在于寻找光滑解。在这个建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）和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.