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

University Of Arkansas, Fayetteville AR

Investigators

Abstract

ABSTRACT -------- The research component of this proposal deals with the study of solutions to certain non-linear systems of Partial Differential Equations 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 models allow 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 of motion under constraints (as in Robot arms motion), and in the study of diffusion in a non isotropic media.