Potential Theory on Metric Measure Spaces

首席研究员建议通过三个项目继续发展度量空间分析的计划。第一个项目将研究与一般度量测量空间上可能不具有黎曼结构的sobolev型函数空间相关的迹空间；这里的想法是找出在给定域的边界上必须确定多少数据，以便获得数据到域内部相应扩展的可靠预测。第二个项目是探索有界几何度量空间中有界域的保形马丁边界。一般来说，共形马丁边界与拉普拉斯算子对应的经典马丁边界不同，它提供了与某些非线性偏微分方程相关的位势理论边界的更可靠的度量。第三个项目是在一般度量空间上构造一个分布导数结构，从而测量域的边界，并确定在这种度量空间中的球体何时是可整流的。在这个项目中提出的研究的潜在应用包括随机过程和抽象度量空间分析之间的联系。抽象度量空间在物理和工程应用中出现，因此本项目中解决的问题也可能对物理和工程产生影响。此外，这些项目将黎曼流形中发散形式的亚椭圆型方程理论与Carnot-Carath 'eodory空间的次黎曼几何中的退化椭圆型方程理论统一起来。同时，保角马丁边界的构造和研究在欧几里得环境下也是一个新的概念，将为研究拟共形映射的边界行为提供新的工具。

The Principal Investigator proposes to continue the program of developing analysis on metric spaces via three projects. The first project will study the trace spaces related to a Sobolev-type function space on general metric measurespaces that may not have a Riemannian structure; the idea here is to find out how well data have to be determined on the boundary of a given domain in order to obtain a reliable prediction of a corresponding extension of the data to theinterior of the domain. The second project is to explore the conformalMartin boundary for bounded domains in metric measure spaces of bounded geometry. The conformal Martin boundary is in general different from the classical Martin boundary corresponding to the Laplacian operator, and provides a more reliable gauge of the potential-theoretic boundary relevant to certain non-linear partial differential equations. The third project is to construct a distributional derivative structure on general metric spaces and thereby measure the boundary of domains and determine when spheres in such a metric space are rectifiable.Potential applications of the research proposed in this project include connections between stochastic processes and analysis in abstract metric spaces. Abstract metric spaces arise in applications inphysics and engineering, and hence the questions addressed in this project havepossible impact in physics and engineering as well. In addition, these projects unify the theory of subelliptic equations of divergence form in Riemannian manifolds with the theory of degenerate elliptic equations in the sub-Riemannian geometry of Carnot-Carath\'eodory spaces. Also, the construction and study of conformal Martin boundary is a new concept even in the Euclidean setting, and will provide a new tool in the study of boundary behavior of quasiconformal mappings.