Potential Theory on Metric Measure Spaces

首席研究员建议通过三个项目继续开展度量空间分析的计划。第一个项目将研究与Sobolev型函数空间有关的迹空间，一般度量测度空间可能没有黎曼结构;这里的想法是找出如何确定给定域边界上的数据，以便获得对域内部数据相应扩展的可靠预测。第二个项目是探讨有界几何的度量测度空间中有界域的共形Martin边界。共形马丁边界一般不同于对应于拉普拉斯算子的经典马丁边界，并且提供了与某些非线性偏微分方程相关的势理论边界的更可靠的规范。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面何时是可求长的。本项目提出的研究的潜在应用包括抽象度量空间中随机过程和分析之间的联系。抽象度量空间在物理学和工程学中有着广泛的应用，因此本课题所研究的问题也可能在物理学和工程学中产生影响。此外，这些项目统一了黎曼流形中的发散形式的次椭圆方程的理论与Carnot-Carath\'eodory空间的次黎曼几何中的退化椭圆方程的理论。同时，共形Martin边界的构造和研究在欧氏空间中也是一个新的概念，为研究拟共形映射的边界性质提供了一个新的工具。

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.