Cacciopoli Sets, Capacities and Quasiconformal Mappings in Carnot-Caratheodory Spaces

本文主要研究Carnot-Caratheodory空间分析中的三个基本问题：第一个问题来自于几何测度论;它与Cacciopoli集及其特征轨迹有关，并应用于极小曲面。第二个问题是从非线性位势理论和亚椭圆型偏微分方程中得到的关于容量的精确估计。第三个问题是关于拟共形映射在Carnot-Caratheodory空间中的可积性和Hausdorff维数偏差。这个理论的根本困难在于这样一个事实，即在卡诺-Caratheodory空间中，我们只能处理有限数量的微分算子（对应于容许的方向），而在欧几里得分析中，所有可能的方向都是允许的。其基本几何结构具有分形特征，这充分说明了上述问题的复杂性和对新思想的需求。由于机器人由机械连杆机构构成，机器人运动必须满足容许方向的无穷小约束。类似的微分约束出现在热力学中描述理想气体阶段的特征量（温度、压力、体积、熵和能量）之间。研究这些以及工程和物理学中相关现象的正确数学背景是卡诺-卡拉西奥多利空间的设定。本建议旨在开发适当的分析工具，以处理这方面的各种实际问题。

We propose to investigate three fundamental problems arising inAnalysis on Carnot-Caratheodory spaces.The first question is from geometric measure theory. It isrelated to Cacciopoli sets and their characteristic locus with applications to minimal surfaces. The second problem is from non-linear potential theory andsub-elliptic PDE related to sharp capacity estimates. The third question concerns the integrability and Hausdorff dimension distortion of quasiconformal mappings in Carnot-Caratheodory spaces. The fundamental difficulty in this theory lies in the fact that in Carnot-Caratheodory spaces we can only work with a restricted number of differential operators (corresponding to admissible directions) while in Euclidean analysis all possible directions were allowed. The underlying geometry has a fractal-like character illustrating well the complexity and the need of new ideas for approaching the above problems.Due to their construction by mechanical linkages, robot motions have to satisfy infinitesimal constrains of admissible directions. Similar differential constrains appear in thermodynamics between the characteristic quantities (temperature, pressure, volume, entropy and energy) describing the stage of an ideal gas. The right mathematical context for studying these and related phenomena from engineering and physics is the setting of Carnot-Caratheodory spaces. The present proposal intends to develop the adequate tools of Analysisfor approaching various practical questions in this context.