Potential Theory on Metric Measure Spaces

度量测度空间的位势理论

• 批准号: 0355027
• 负责人: Nageswari Shanmugalingam
• 金额: $ 9.85万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355027&HistoricalAwards=false
• 关键词: Potential; Theory; Metric; Measure; Spaces

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.

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可能不具有黎曼结构的一般度量测度空间上的索博列夫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数据的程度，以便获得数据到域内部的相应扩展的可靠预测。第二个项目是探索有界几何的度量测量空间中的有界域的共角马丁边界。共形马丁边界通常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建分布导数结构，从而测量域的边界并确定此类度量空间中的球体何时可校正。该项目中提出的研究的潜在应用包括随机过程与分析之间的联系抽象的度量空间。抽象度量空间出现在物理学和工程学的应用中，因此该项目解决的问题也可能对物理学和工程学产生影响。此外，这些项目将黎曼流形中发散形式的次椭圆方程理论与卡诺-卡拉特空间次黎曼几何中的简并椭圆方程理论统一起来。此外，即使在欧几里得环境中，共形马丁边界的构建和研究也是一个新概念，将为研究拟共形映射的边界行为提供新的工具。