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Potential Theory of Functions of Bounded Variation and Quasiconformal Maps

有界变分函数和拟共形映射的势理论

基本信息

批准号：
1500440
负责人：
Nageswari Shanmugalingam
金额：
$ 29.78万
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：
2015
资助国家：
美国
起止时间：
2015-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500440&HistoricalAwards=false
关键词：
Potential Theory Functions Bounded Variation

项目摘要

When we view images of objects, what we see are various locations of the object emitting different intensities of light. We can think of the image of the object as formed by a collection of surfaces of different shapes, each with its own uniform brightness. Similarly, in the study of various mathematical objects such as functions that measure temperature at different locations, functions that measure electro-magnetic intensities, and functions that measure the velocity of fluid particles at various places in a fluid, can be understood in terms of the shape of the level sets of the function. (A level set is a set where the function takes on a given constant value.) This project in metric space analysis explores the behavior of functions that arise in the study of potential theory and quasiconformal mappings in terms of level sets. Applications of the the research project include image processing and edge detection. Many components of this project are suitable dissertation material for graduate students, and therefore will contribute to the training of future members of the STEM workforce.This project is concerned with links between geometry of a metric space given in terms of its sets of finite perimeter on the one hand, and nonlinear potential theory and quasiconformal mappings on the other hand. The spaces considered are equipped with a doubling measure supporting a 1-Poincare inequality. In the first part of the project the PI will explore interactions between collections of sets of finite perimeter and quasiconformal mappings, and between collections of sets of finite perimeter and nonlinear potential theory. The second part of the project will explore "tangent space" regularity of sets of quasiminimal boundary surfaces. The third part of the project is to develop a potential theory for functions of bounded variation. The last part of the project is to obtain a characterization of certain Poincare inequalities in terms of modulus of families of sets of finite perimeter. Applications of the research include further understanding of connections between metric geometry related to sets of finite perimeter and solutions to certain nonlinear partial differential equations. Such sets arise in the study of image processing and edge detection, while abstract metric spaces arise in Riemannian manifolds theory when considering Gromov-Hausdorff limit spaces as found in the works of Cheeger, Gromov, and Perelman. Furthermore, these projects will expand the current knowledge about geometry and the theory of sets of finite perimeter in Carnot-Caratheodory spaces.

当我们查看物体的图像时，我们看到的是物体的不同位置发出不同强度的光。我们可以认为物体的图像是由不同形状的表面的集合形成的，每个表面都有自己均匀的亮度。类似地，在研究各种数学对象时，例如测量不同位置的温度的函数、测量电磁强度的函数以及测量流体中不同位置的流体粒子的速度的函数，可以根据函数的水平集的形状来理解。（水平集是函数取给定常数值的集合。）度量空间分析中的这个项目探讨了在水平集方面的势论和拟共形映射研究中出现的函数的行为。该研究项目的应用包括图像处理和边缘检测。该项目的许多组成部分都是适合研究生的论文材料，因此将有助于培训 STEM 劳动力的未来成员。该项目一方面关注以有限周长集给出的度量空间的几何形状，另一方面关注非线性势论和拟共形映射之间的联系。所考虑的空间配备了支持 1-庞加莱不等式的加倍测度。在该项目的第一部分中，PI 将探索有限周长集合和拟共形映射之间的相互作用，以及有限周长集合和非线性势理论之间的相互作用。该项目的第二部分将探索准最小边界面组的“切线空间”规律。该项目的第三部分是开发有界变分函数的潜在理论。该项目的最后一部分是根据有限周长集族的模数来获得某些庞加莱不等式的表征。该研究的应用包括进一步理解与有限周长组相关的度量几何与某些非线性偏微分方程的解之间的联系。此类集合出现在图像处理和边缘检测的研究中，而抽象度量空间出现在黎曼流形理论中，当考虑 Cheeger、Gromov 和 Perelman 的著作中发现的 Gromov-Hausdorff 极限空间时。此外，这些项目将扩展当前关于几何学和卡诺-卡拉西奥多里空间中有限周长集理论的知识。