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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.

当我们观察物体的图像时，我们看到的是物体发出不同强度光的不同位置。我们可以认为物体的图像是由不同形状的表面组成的，每个表面都有自己统一的亮度。同样，在研究各种数学对象时，例如测量不同位置温度的函数、测量电磁强度的函数以及测量流体中不同位置流体粒子速度的函数，都可以根据形状来理解。函数的水平集。（水平集是函数取给定常数值的集合。这个度量空间分析的项目探讨了在势理论和拟共形映射的研究中出现的函数在水平集方面的行为。本研究计画的应用领域包括影像处理与边缘侦测.本项目的许多内容都是适合研究生的论文材料，因此将有助于培养未来的STEM工作人员。本项目关注的是度量空间的几何之间的联系，一方面是它的有限周长集，另一方面是非线性势理论和拟共形映射。考虑的空间配备了一个加倍措施支持1-庞加莱不等式。在该项目的第一部分，PI将探讨有限周长集和拟共形映射之间的相互作用，以及有限周长集和非线性势理论之间的相互作用。第二部分的项目将探讨“切空间”的正则性集的拟最小边界曲面。本项目的第三部分是发展有界变差函数的势理论。该项目的最后一部分是获得某些庞加莱不等式的模的有限周长的集合的家庭的一个特征。研究的应用包括进一步理解与有限周长集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。这样的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，当考虑在Cheeger，Gromov和Perelman的作品中发现的Gromov-Hausdorff极限空间时。此外，这些项目将扩大目前的知识几何和理论的集合有限周长在卡诺-Caratheodory空间。